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 Perl (v5.18.2) (judge verdict: Time Limit Exceeded)

use v5.18;
use warnings;
sub read_line { chomp(my $s = <>); $s }
sub splited_read_line { split / /, read_line }
 
my $n = read_line;
my @balls = splited_read_line;
say solve(\@balls);
 
sub solve {
  my $balls = shift;
  my @sorted_balls = sort { $b <=> $a } @$balls;
 
  my $count = 0;
  for (
    my $ball = $sorted_balls[my $i = 0];
    $i < $#sorted_balls;
    $ball = $sorted_balls[++$i]
  ) {
    next unless defined $ball;
    for (
      my $compare_ball = $sorted_balls[my $j = $i + 1];
      $j < @sorted_balls;
      $compare_ball = $sorted_balls[++$j]
    ) {
      next unless defined $compare_ball;
      my $sum = $compare_ball + $ball;
      if ( ( log($sum) / log(2) ) =~ /^[0-9]+$/ ) {
        $count++;
        $sorted_balls[$j] = undef;
        last;
      }
    }
    $sorted_balls[$i] = undef;
  }
  $count;
}