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 PYTHON3):

Solution:
# Manhattan Distance
distance = (
    (),
    (0, 1, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 6),
    (1, 0, 1, 2, 2, 1, 2, 3, 3, 2, 3, 4, 4, 3, 4, 5),
    (2, 1, 0, 1, 3, 2, 1, 2, 4, 3, 2, 3, 5, 4, 3, 4),
    (3, 2, 1, 0, 4, 3, 2, 1, 5, 4, 3, 2, 6, 5, 4, 3),
    (1, 2, 3, 4, 0, 1, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5),
    (2, 1, 2, 3, 1, 0, 1, 2, 2, 1, 2, 3, 3, 2, 3, 4),
    (3, 2, 1, 2, 2, 1, 0, 1, 3, 2, 1, 2, 4, 3, 2, 3),
    (4, 3, 2, 1, 3, 2, 1, 0, 4, 3, 2, 1, 5, 4, 3, 2),
    (2, 3, 4, 5, 1, 2, 3, 4, 0, 1, 2, 3, 1, 2, 3, 4),
    (3, 2, 3, 4, 2, 1, 2, 3, 1, 0, 1, 2, 2, 1, 2, 3),
    (4, 3, 2, 3, 3, 2, 1, 2, 2, 1, 0, 1, 3, 2, 1, 2),
    (5, 4, 3, 2, 4, 3, 2, 1, 3, 2, 1, 0, 4, 3, 2, 1),
    (3, 4, 5, 6, 2, 3, 4, 5, 1, 2, 3, 4, 0, 1, 2, 3),
    (4, 3, 4, 5, 3, 2, 3, 4, 2, 1, 2, 3, 1, 0, 1, 2),
    (5, 4, 3, 4, 4, 3, 2, 3, 3, 2, 1, 2, 2, 1, 0, 1)
)
 
def get_distance(board):
    sd = 0
    for i in range(16):
        p = board[i]
        if p == 0:
            continue
        sd += distance[p][i]
    return sd
 
adjacent = (
    (1, 4),         # 0
    (0, 2, 5),      # 1
    (1, 3, 6),      # 2
    (2, 7),         # 3
    (0, 5, 8),      # 4
    (1, 4, 6, 9),   # 5
    (2, 5, 7, 10),  # 6
    (3, 6, 11),     # 7
    (4, 9, 12),     # 8
    (5, 8, 10, 13), # 9
    (6, 9, 11, 14), # 10
    (7, 10, 15),    # 11
    (8, 13),        # 12
    (9, 12, 14),    # 13
    (10, 13, 15),   # 14
    (11, 14)        # 15
)
 
GOAL = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0]
 
import sys
import heapq
 
# A* algorithm
def solve():
    board = list(map(int, sys.stdin.read().split()))
    h_cost = get_distance(board)
    state = (h_cost, board)
     
    q = [state] # priority queue
    d = {tuple(board): True}
     
    while q:
        state1 = heapq.heappop(q)
        h_cost, board = state1
        if board == GOAL:
            return h_cost
        space = board.index(0)
        for i in adjacent[space]:
            p = board[i]
            new_board = board[:]
            new_board[space], new_board[i] = p, 0
            new_h_cost = h_cost + 1 - distance[p][i] + distance[p][space]
 
            key = tuple(new_board)
            if key not in d and new_h_cost <= 45:
                new_state = (new_h_cost, new_board)
                heapq.heappush(q, new_state)
                d[key] = True
 
print(solve())