Problem:
Score : 600 points

Problem Statement

Takahashi has N balls with positive integers written on them. The integer written on the i-th ball is A_i.
He would like to form some number of pairs such that the sum of the integers written on each pair of balls is a power of 2.
Note that a ball cannot belong to multiple pairs.
Find the maximum possible number of pairs that can be formed.

Here, a positive integer is said to be a power of 2 when it can be written as 2^t using some non-negative integer t.

Constraints

1 \leq N \leq 2\times 10^5

1 \leq A_i \leq 10^9

A_i is an integer.

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 ... A_N

Output

Print the maximum possible number of pairs such that the sum of the integers written on each pair of balls is a power of 2.

Sample Input 1

3
1 2 3

Sample Output 1

1

We can form one pair whose sum of the written numbers is 4 by pairing the first and third balls.
Note that we cannot pair the second ball with itself.

Sample Input 2

5
3 11 14 5 13

Sample Output 2

2

Solution in PyPy3 (7.3.0) (judge verdict: Accepted)

import sys

readline = sys.stdin.readline
MOD = 10 ** 9 + 7
INF = float('INF')
sys.setrecursionlimit(10 ** 5)


def main():
    from collections import Counter

    n = int(readline())
    a = list(map(int, readline().split()))
    counter = Counter(a)

    keys = list(counter.keys())
    keys.sort(reverse=True)

    ans = 0

    for num1 in keys:
        c1 = counter[num1]
        x = 1
        while x <= num1:
            x *= 2
        num2 = x - num1
        if num1 == num2:
            cnt = c1 // 2
            ans += cnt
            counter[num1] -= cnt * 2
        else:
            c2 = counter[num2]
            cnt = min(c1, c2)
            ans += cnt
            counter[num1] -= cnt
            counter[num2] -= cnt

    print(ans)


if __name__ == '__main__':
    main()