Problem: Allen and Bessie are playing a simple number game. They both know a function f: \{0, 1\}^n → R, i. e. the function takes n binary arguments and returns a real value. At the start of the game, the variables x_1, x_2, ..., x_n are all set to -1. Each round, with equal probability, one of Allen or Bessie gets to make a move. A move consists of picking an i such that x_i = -1 and either setting x_i → 0 or x_i → 1.

After n rounds all variables are set, and the game value resolves to f(x_1, x_2, ..., x_n). Allen wants to maximize the game value, and Bessie wants to minimize it.

Your goal is to help Allen and Bessie find the expected game value! They will play r+1 times though, so between each game, exactly one value of f changes. In other words, between rounds i and i+1 for 1 ≤ i ≤ r, f(z_1, ..., z_n) → g_i for some (z_1, ..., z_n) ∈ \{0, 1\}^n. You are to find the expected game value in the beginning and after each change.

Input

The first line contains two integers n and r (1 ≤ n ≤ 18, 0 ≤ r ≤ 2^{18}).

The next line contains 2^n integers c_0, c_1, ..., c_{2^n-1} (0 ≤ c_i ≤ 10^9), denoting the initial values of f. More specifically, f(x_0, x_1, ..., x_{n-1}) = c_x, if x = \overline{x_{n-1} … x_0} in binary.

Each of the next r lines contains two integers z and g (0 ≤ z ≤ 2^n - 1, 0 ≤ g ≤ 10^9). If z = \overline{z_{n-1} ... z_0} in binary, then this means to set f(z_0, ..., z_{n-1}) → g.

Output

Print r+1 lines, the i-th of which denotes the value of the game f during the i-th round. Your answer must have absolute or relative error within 10^{-6}.

Formally, let your answer be a, and the jury's answer be b. Your answer is considered correct if \frac{|a - b|}{max{(1, |b|)}} ≤ 10^{-6}.

Examples

Input

2 2
0 1 2 3
2 5
0 4


Output

1.500000
2.250000
3.250000


Input

1 0
2 3


Output

2.500000


Input

2 0
1 1 1 1


Output

1.000000

Note

Consider the second test case. If Allen goes first, he will set x_1 → 1, so the final value will be 3. If Bessie goes first, then she will set x_1 → 0 so the final value will be 2. Thus the answer is 2.5.

In the third test case, the game value will always be 1 regardless of Allen and Bessie's play.

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
int main(int argc, char **argv) {
  int n, r;
  std::cin >> n >> r;
  int s = pow(2, n);
  std::vector<uint64_t> values(s);
  for (int i = 0; i < values.size(); ++i) {
    std::cin >> values[i];
  }
  uint64_t sum = std::accumulate(values.begin(), values.end(), (uint64_t)0);
  double avg = (double)sum / s;
  std::cout << std::setprecision(6) << avg << '\n';
  for (int i = 1; i <= r; ++i) {
    int index;
    uint64_t new_val;
    std::cin >> index >> new_val;
    uint64_t current_val = values[index];
    if (new_val != current_val) {
      uint64_t diff = new_val - current_val;
      uint64_t new_sum = sum + diff;
      values[index] = new_val;
      avg = (double)new_sum / s;
      sum = new_sum;
    }
    if (avg > 1000000.0) {
      uint64_t avg_rounded = (uint64_t)avg;
      std::cout << avg_rounded << '\n';
    } else {
      std::cout << std::setprecision(17) << avg << "\n";
    }
  }
}