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 <cstdio>
#include <iostream>
#include <cmath>
#include <string>
#include <cassert>
#include <algorithm>
using namespace std;

static const int N = 4;
static const int N2 = N*N;
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];
  int space;
  int 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 dfs(int depth, int prev){
  if(state.MD == 0) return true;
  //現在の深さにヒューリスティック距離を足して制限を超えたら枝をかる
  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_deeping(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(){
  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_deeping(in);
  cout << ans.size() << endl;

  return 0;
}