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 (2.4.0) (judge verdict: Accepted)

import bisect
import math
import sys
input = sys.stdin.readline


N = int(input())
A = list(map(int, input().split()))
A.sort()
A_set = []
num = []
a_cur = float('inf')
for a in A:
    if a != a_cur:
        A_set.append(a)
        num.append(1)
        a_cur = a
    else:
        num[-1] += 1

ans = 0
for i in range(len(A_set) - 1, -1, -1):
    if num[i] > 0:
        A_i = A_set[i]
        comp = 2 ** math.ceil(math.log(A_i + 1, 2)) - A_i
        j = bisect.bisect_left(A_set, comp, 0, i)
        while True:
            if A_set[j] == comp:
                if num[j] > 0:
                    if i != j:
                        ans += min(num[j], num[i])
                        num[j] = max(num[j] - num[i], 0)
                        break
                    elif num[j] >= 2:
                        add = num[j] // 2
                        ans += add
                        num[j] = num[j] % 2
                        break
                    else:
                        break
                else:
                    break
            else:
                break

print(ans)