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: Wrong Answer)

#!/usr/bin/perl

use strict;
use warnings;
use 5.010;

use Data::Dumper;

my @INPUT;
while (<>) {
    chomp;
    push @INPUT, $_;
}

my $N = shift @INPUT;
my @A = split " ", shift @INPUT;
my %A;
for (my $i = 0; $i <= $#A; $i++) {
    $A{$A[$i]} = $i;
}

my @twot;
for (my $i = 0; 2 ** $i < 10 ** 9; $i++) {
    push @twot, 2 ** $i;
}

my $ans = 0;
my $min = 1;
while ($min < 3) {
    my $found = hoge($min);
    if ($found == 1) {
        $min = 1;
    } elsif ($found == 100000) {
        last;
    } else {
        $min += $found;
    }
}

sub hoge {
    my $min = shift;
    #say "min: $min";
    my $min_cnt = 100000;
    for (my $i = 0; $i < @A; $i++) {
        my $a = $A[$i];
        #say "i: $a";
        if ($a == 0) {
            next;
        }
        my $cnt = 0;
        my $pair;
        for my $twot (reverse @twot) {
            my $pair_v = $twot - $a;
            if ($pair_v <= 0) {
                last;
            }
            if (defined $A{$pair_v} && $A{$pair_v} != $i) {
                $pair = $A{$pair_v};
                $cnt++;
            }
        }
        if ($cnt && $min_cnt > $cnt) {
            $min_cnt = $cnt;
        }
        #say "cnt: $cnt";
        if ($cnt == $min) {
            $ans++;
            #say "$A[$i] + $A[$pair]";
            delete $A{$A[$pair]};
            $A[$i] = 0;
            $A[$pair] = 0;
            return 1;
        }
    }
    return $min_cnt;
}
say $ans;


__END__

my %pairs;
my $cnt = 0;
for (my $i = 0; $i < @A; $i++) {
    my $a = $A[$i];
    for my $twot (reverse @twot) {
        my $pair = $twot - $a;
        if ($pair <= 0) {
            last;
        }
        if (defined $A{$pair} && $A{$pair} != $i) {
            push @{$pairs{$i}}, $A{$pair};
            $cnt++;
        }
    }
}
print Dumper \%pairs;

my %num_of_pair;
for (my $i = 0; $i < @A; $i++) {
    next unless defined $pairs{$i};
    $num_of_pair{$i} = scalar @{$pairs{$i}};
}