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<cassert>
#include<string>
#include<map>
#include<queue>

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], 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 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;
}

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

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;
        }
    }

}
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_deepening(in);
    cout << ans.size() << endl;
    return 0;
}