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:
from heapq import heappop, heappush

manhattan = [[abs((i % 4) - (j % 4)) + abs((i // 4) - (j // 4)) for j in range(16)] for i in range(16)]
movables = [{1, 4}, {0, 2, 5}, {1, 3, 6}, {2, 7}, {0, 5, 8}, {1, 4, 6, 9}, {2, 5, 7, 10}, {3, 6, 11},
            {4, 9, 12}, {5, 8, 10, 13}, {6, 9, 11, 14}, {7, 10, 15}, {8, 13}, {9, 12, 14}, {10, 13, 15}, {11, 14}]
swap_cache = [[(1 << mf) - (1 << mt) for mt in range(0, 64, 4)] for mf in range(0, 64, 4)]
destination = 0xfedcba9876543210


def swap(board, move_from, move_to):
    return board + swap_cache[move_from][move_to] * (15 - ((board >> (4 * move_from)) & 15))


i = 0
board_init = 0
blank_init = 0
for _ in range(4):
    for n in map(int, input().split()):
        if n:
            n -= 1
        else:
            n = 15
            blank_init = i
        board_init += n * 16 ** i
        i += 1

estimation_init = sum(manhattan[i][((board_init >> (4 * i)) & 15)] for i in range(16) if i != blank_init)

queue = [(estimation_init, board_init, blank_init)]
visited = set()

while True:
    estimation, board, blank = heappop(queue)
    if board in visited:
        continue
    elif board == destination:
        print(estimation)
        break
    visited.add(board)
    for new_blank in movables[blank]:
        new_board = swap(board, new_blank, blank)
        if new_board in visited:
            continue
        num = (board >> (4 * new_blank)) & 15
        new_estimation = estimation + 1 - manhattan[new_blank][num] + manhattan[blank][num]
        if new_estimation > 45:
            continue
        heappush(queue, (new_estimation, new_board, new_blank))