Problem: You are given a set $T$, which is a subset of $S$. The set $S$ consists of $0, 1, ... n-1$. Print all subsets of $T$. Note that we represent $0, 1, ... n-1$ as 00...0001, 00...0010, 00...0100, ..., 10...0000 in binary respectively and the integer representation of a subset is calculated by bitwise OR of existing elements.

Constraints

* $1 \leq n \leq 28$
* $0 \leq k \leq 18$
* $k \leq n$
* $0 \leq b_i < n$

Input

The input is given in the following format.


$n$
$k \; b_0 \; b_1 \; ... \; b_{k-1}$


$k$ is the number of elements in $T$, and $b_i$ represents elements in $T$.

Output

Print the subsets ordered by their decimal integers. Print a subset in the following format.


$d$: $e_0$ $e_1$ ...


Print ':' after the integer value $d$, then print elements $e_i$ in the subset in ascending order. Separate two adjacency elements by a space character.

Example

Input

4
2 0 2


Output

0:
1: 0
4: 2
5: 0 2

Solution (in PYTHON3):

Solution:
import itertools



n = int(input())

k, *list_b = map(int, input().split())

list_d = []

for k in range(len(list_b) + 1):

    for tuple_e in itertools.combinations(list_b, k):
        d = 0

        for bit in tuple_e:
            d += 2 ** bit

        list_d.append((d, tuple_e))


for d, tuple_e in sorted(list_d):
    if tuple_e:
        print(f"{d}: ", end="")
        print(*tuple_e)
    else:
        print(f"{d}:")