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 Ruby (2.3.3) (judge verdict: Runtime Error)

def solve(ai)
  ai.sort!
  t1 = Math.log2(ai[0]).ceil
  t2 = Math.log2(ai[-1]).floor
  pp t1,t2,ai

  def find_matching(a,counts)
    i = 0
    j = a.size-1
    while a[i] <= 0 and a[j] >= 0 and i < j
      # pp "i,j,#{a[i]},#{a[j]}"
      if -a[i] == a[j]
        counts[i] += 1
        counts[j] += 1
        i += 1
        j -= 1
      elsif -a[i] > a[j]
        i += 1
      elsif -a[i] < a[j]
        j -= 1
      end
    end
  end


  counts = Array.new(ai.size,0)
  (t1..t2).each do |t|
    x = 2**t
    mapped = ai.map {|a| a-x }
    find_matching(mapped, counts)
    pp "mapped", mapped, "counts:", counts
  end

  # pp counts
  i = 0
  num_matching = 0
  while i < counts.size
    n = counts[i]
    num_a = 1
    while ai[i] == ai[i+1]
      i += 1
      n += counts[i]
      num_a += 1
    end
    i += 1
    num_matching += (num_a<=n) ? num_a : n
  end

  num_matching/2
end

if __FILE__ == $0
  N = readline.to_i
  ai = readline.split.map(&:to_i)

  puts solve(ai)
end
# puts (counts.count {|x| x>0})/2