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<iostream>
#include<cmath>
#include<string>
#include<map>
#include<queue>

using namespace std;
#define N 4
#define N2 16

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

  bool operator < (const Puzzle &p) const {
    for(int i=0;i<N2;i++) {
      if(f[i]==p.f[i])continue;
      return f[i] < p.f[i];
    }
    return false;
  }
};

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

struct State {
  Puzzle puzzle;
  int estimated;
  bool operator < (const State &s) const {
    return estimated > s.estimated;
  }
};

int getAIIMD(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;
}

int astar(Puzzle s) {
  priority_queue<State> PQ;
  s.MD = getAIIMD(s);
  s.cost=0;
  map<Puzzle, bool> V;
  Puzzle u,v;
  State initial;
  initial.puzzle = s;
   initial.estimated = getAIIMD(s);
   PQ.push(initial);

   while(!PQ.empty()) {
     State st = PQ.top(); PQ.pop();
     u=st.puzzle;

     if(u.MD == 0) return u.cost;
     V[u] = true;

     int sx = u.space/N;
     int sy = u.space%N;

     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;
       v = u;

       v.MD -= MDT[tx * N + ty][v.f[tx * N + ty] -1];
       v.MD += MDT[sx * N + sy][v.f[tx * N +ty] - 1];

       swap(v.f[sx * N + sy],v.f[tx * N + ty]);
       v.space  = tx * N +ty;
       if(!V[v]) {
         v.cost++;
         State news;
         news.puzzle = v;
         news.estimated = v.cost + v.MD;
         PQ.push(news);
       }
     }
   }
   return -1;
}

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;
   }
 }
 cout << astar(in) << endl;

 return 0;
}