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:
// 15 puzzle

#include<iostream>
#include<cmath>
#include<string>
#include<map>
#include<queue>
#include<cassert>
 
using namespace std;
 
#define N 4
#define N2 16
#define 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);
bool isSolved();
bool dfs(int depth,int prev);
string iterative_deepening(Puzzle in);
 
int main(void)
{
  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;
}
 
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";
}