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:
Truth table for Sheffer stroke function
x y x|y
0 0 1
0 1 1
1 0 1
1 1 0
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:
Ai is an expression that denotes value of ith bit of number A.
Bi is 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.
Input
The input contains a single integer N (1 <= N <= 100).
Output
Write to the output 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 shall not exceed 50*N bytes.
This problem contains multiple test cases!
The first line of a multiple input is an integer N, then a blank line followed by N input blocks. Each input block is in the format indicated in the problem description. There is a blank line between input blocks.
The output format consists of N output blocks. There is a blank line between output blocks.
Sample Input
1
2
Sample Output
((A1|B1)|(((A0|B0)|(A0|B0))|((A1|A1)|(B1|B1))))
