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 LuaJIT (2.0.4) (judge verdict: Accepted)

local mce, mfl, msq, mmi, mma = math.ceil, math.floor, math.sqrt, math.min, math.max
local function lower_bound(ary, x, num)
  if num == 0 then return 1 end
  if(x <= ary[1].val) then return 1 end
  if(ary[num].val < x) then return num + 1 end
  local min, max = 1, num
  while(1 < max - min) do
    local mid = mfl((min + max) / 2)
    if(ary[mid].val < x) then
      min = mid
    else
      max = mid
    end
  end
  return max
end
local function upper_bound(ary, x, num)
  if num == 0 then return 1 end
  if(x < ary[1].val) then return 1 end
  if(ary[num].val <= x) then return num + 1 end
  local min, max = 1, num
  while(1 < max - min) do
    local mid = mfl((min + max) / 2)
    if(ary[mid].val <= x) then
      min = mid
    else
      max = mid
    end
  end
  return max
end
local n = io.read("*n", "*l")
local t = {}
local str = io.read()
for v in str:gmatch("%d+") do
  local tmp = {}
  tmp.val = tonumber(v)
  tmp.used = false
  table.insert(t, tmp)
end
table.sort(t, function(x, y) return x.val < y.val end)
local cnt = 0
for i = n, 2, -1 do
  if not t[i].used then
    t[i].used = true
    local v = t[i].val
    local mul = 2
    while mul <= v do
      mul = mul * 2
    end
    v = mul - v
    local lb, ub = lower_bound(t, v, i - 1), upper_bound(t, v, i - 1)
    for j = ub - 1, lb, -1 do
      if not t[j].used then
        cnt = cnt + 1
        t[j].used = true
        break
      end
    end
  end
end
print(cnt)