標籤：

世界上最大的數字和最小的數字分別是多少？

01-24

你是不是想問世界上誰最有錢，誰跑得最快，誰蓋的樓最高，誰長得最胖對吧？譬如說最小的溫度，-273度就是最小的，不可能存在比這個更低的溫度了。譬如速度，最快的光速為299792458m/s,不存在比這個更快的速度了。

0。0中包含著負無窮大與正無窮大。

沒有最大的數的。

就算有一個最大數，那我也能舉例出一個最大數+1，從而推翻前面的最大數。

誒~我這麼說你明白了吧

如果「世界」是一個物理概念的話，那我想題主有可能問的是「在現有的剛量體系下，宇宙中有意義的、最大的數量級和最小的數量級」是多少，比如星體總數，夸克質量之類的。

還是有討論價值的，如果好好問問題的話。

這是個哲學題

知乎現在普及到小學生了嗎？

只有無窮大∞和無窮小這兩個概念

我記得數字和數不是同一個概念。

求求你買一本高等數學教材，隨便看一本就好了

可以去了解一下數學家Georg Cantor呀：）

對不起您走錯了，想知道請出門右轉百度知道

如果你說的是指實數域上的數的話，我認為這個問題應該等價於問是否存在最大的正整數

答案是不存在的，建議百度一下皮亞諾公理

任取一個正整數q，都存在一個後繼數q』

從而可以證明這個q不是最大的

然後由q的任意性就可以說明不存在最大的正整數

busy beaver函數一個比一個大，第七個以上還沒人算出來過。。。

目前用到的比較有建樹證明裡面，最大的有限數應該就是Graham number了吧

舉個栗子怎麼算吧

a↑b=a*a*a

3↑3不過就是3^3，好簡單的。

但是接下來就大了

a↑↑b=a↑(a↑……↑a)

a出現b次

3↑↑3=3↑(3↑3)

也就是3^3^3=7625597484987

a↑↑↑b=a↑↑(a↑↑……↑↑a)

a出現b次

3↑↑↑3=3↑↑(3↑↑3)

也就是把3給3次方大概7600000000000次吧。

後面的以此類推。

不過一般遞歸演算法往大了走的話都能挺大的

(所以有比我還閑的人可以比一比這個大還是tree(3)大，雖然一點也不難，可是好無聊啊)

但是！你要說正實數的最大和最小的話，那應該是沒有的

通過阿基米德性應該就能證明了

對於任意兩個實數x，y，x&>0，y&>0，總有一整數N使得Nx&>y

你隨便找個y，我就能有個Nx比它大

你隨便找個x，我就能有個y/N(如果N是負數那就找-y/N)比x小

99999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧999999999999999999999999∧9999999999999

推薦閱讀：

※關於愛德華威滕的物理學造詣？
※2的67次方減1是質數嗎，這個問題是怎麼回事？是怎樣一個故事？
※遞推數列通項的一些解法
※矩陣論什麼好的書籍推薦？
※在任意的二維共形場論里，中心荷一定是實數嗎？

TAG:數學 |

分頁阅读： 1 2