Problem: The goal of the 15 puzzle problem is to complete pieces on $4 \times 4$ cells where one of the cells is empty space.

In this problem, the space is represented by 0 and pieces are represented by integers from 1 to 15 as shown below.


1 2 3 4
6 7 8 0
5 10 11 12
9 13 14 15


You can move a piece toward the empty space at one step. Your goal is to make the pieces the following configuration in the shortest move (fewest steps).


1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 0


Write a program which reads an initial state of the puzzle and prints the fewest steps to solve the puzzle.

Constraints

* The given puzzle is solvable in at most 45 steps.

Input

The $4 \times 4$ integers denoting the pieces or space are given.

Output

Print the fewest steps in a line.

Example

Input

1 2 3 4
6 7 8 0
5 10 11 12
9 13 14 15


Output

8

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
using namespace std;

static const int N = 4;
static const int N2 = 16;
static const int LIMIT = 100;

static const int dx[4] = { 0, -1,  0, 1};
static const int dy[4] = { 1,  0, -1, 0};
static const char dir[4] = {'r', 'u', 'l', 'd'};
int MDT[N2][N2];

struct Puzzle {
  int f[N2], space, MD;
};

Puzzle state;
int limit;
int path[LIMIT];

int getAllMD(Puzzle pz) {
  int sum = 0;
  for (int i = 0; i < N2; i++) {
    if (pz.f[i] == N2) continue;
    sum += MDT[i][pz.f[i] - 1];
  }
  return sum;
}

bool isSolved() {
  for (int i = 0; i < N2; i++) {
    if (state.f[i] != i+1) return false;
  }
  return true;
}

bool dfs(int depth, int prev) {
  if (state.MD == 0) return true;
  if (depth + state.MD > limit) return false;

  int sx = state.space / N;
  int sy = state.space % N;
  Puzzle tmp;

  for (int r = 0; r < 4; r++) {
    int tx = sx + dx[r];
    int ty = sy + dy[r];
    if (tx < 0 || ty < 0 || tx >= N || ty >= N) continue;
    if ((max(prev, r) - min(prev, r)) == 2) continue;

    tmp = state;
    state.MD -= MDT[tx*N+ty][state.f[tx*N+ty]-1];
    state.MD += MDT[sx*N+sy][state.f[tx*N+ty]-1];
    swap(state.f[tx*N+ty], state.f[sx*N+sy]);
    state.space = tx*N+ty;
    if (dfs(depth+1, r)) {
      path[depth] = r;
      return true;
    }
    state = tmp;
  }
  return false;
}

string iterative_deepening(Puzzle in) {
  in.MD = getAllMD(in);

  for (limit = in.MD; limit <= LIMIT; limit++) {
    state = in;
    if (dfs(0, -100)) {
      string ans = "";
      for (int i = 0; i < limit; i++) ans += dir[path[i]];
      return ans;
    }
  }
  return "Unsolvable";
}

int main(void) {
    cin.tie(0);
    ios::sync_with_stdio(false);

    for (int i = 0; i < N2; i++) {
      for (int j = 0; j < N2; j++) {
        MDT[i][j] = abs(i/N - j/N) + abs(i%N - j%N);
      }
    }

    Puzzle in;
    for (int i = 0; i < N2; i++) {
      cin >> in.f[i];
      if (in.f[i] == 0) {
        in.f[i] = N2;
        in.space = i;
      }
    }

    string ans = iterative_deepening(in);
    cout << ans.size() << endl;

    return 0;
}