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It is known that Sheffer stroke function (NOT-AND) can be used to construct any Boolean function. The truth table for this function is given below:
Consider the problem of adding two binary numbers A and B, each containing N bits. The individual bits of A and B are numbered from 0 (the least significant) to N-1 (the most significant). The sum of A and B can always be represented by N+1 bits. Let's call most significant bit of the sum (bit number N) the overflow bit.
Your task is to construct a logical expression using the Sheffer stroke function that computes the value of the overflow bit for arbitrary values of A and B. Your expression shall be constructed according to the following rules:
Aiis an expression that denotes value of ith bit of number A.
Biis an expression that denotes value of ith bit of number B.
(x|y)is an expression that denotes the result of Sheffer stroke function for x and y, where x and y are expressions.
When writing the index, i, for bits in A and B, the index shall be written as a decimal number without leading zeros. For example, bit number 12 of A must be written as
A12. The expression should be completely parenthesized (according to the 3rd rule). No blanks are allowed inside the expression.
The input file contains a single integer N (1 ≤ N ≤ 100).
Write to the output file an expression for calculating overflow bit of the addition of two N-bit numbers A and B according to the rules given in the problem statement.
Note: The stroke symbol ( | ) is an ASCII character with code 124 (decimal).
The output file size shall not exceed 50*N bytes.