Problem:
Score : 500 points

Problem Statement

Given are N integers A_1,\ldots,A_N.

We will choose exactly K of these elements. Find the maximum possible product of the chosen elements.

Then, print the maximum product modulo (10^9+7), using an integer between 0 and 10^9+6 (inclusive).

Constraints

1 \leq K \leq N \leq 2\times 10^5

|A_i| \leq 10^9

Input

Input is given from Standard Input in the following format:

N K
A_1 \ldots A_N

Output

Print the maximum product modulo (10^9+7), using an integer between 0 and 10^9+6 (inclusive).

Sample Input 1

4 2
1 2 -3 -4

Sample Output 1

12

The possible products of the two chosen elements are 2, -3, -4, -6, -8, and 12, so the maximum product is 12.

Sample Input 2

4 3
-1 -2 -3 -4

Sample Output 2

1000000001

The possible products of the three chosen elements are -24, -12, -8, and -6, so the maximum product is -6.

We print this value modulo (10^9+7), that is, 1000000001.

Sample Input 3

2 1
-1 1000000000

Sample Output 3

1000000000

The possible products of the one chosen element are -1 and 1000000000, so the maximum product is 1000000000.

Sample Input 4

10 10
1000000000 100000000 10000000 1000000 100000 10000 1000 100 10 1

Sample Output 4

999983200

Be sure to print the product modulo (10^9+7).

Solution in Haskell (GHC 8.8.3) (judge verdict: Time Limit Exceeded)

{-# LANGUAGE TypeApplications #-}

module Main where

import Control.Monad (filterM)
import Data.List (partition, sort, sortOn)
import Data.Ord (Down(Down))
import Data.Bifunctor (Bifunctor(bimap))

main :: IO ()
main = do
  (n : k : _) <- map (read @Int) . words <$> getLine
  (ps, ns) <- bimap (take k . sortOn Down) sort . partition (>= 0) . map read . words <$> getLine
  let as = filter ((==) k . length) $ powerset (ps ++ ns)
  let maxP = maximum [ product p | p <- as ]
  print $ maxP `mod` 1000000007
  where
    powerset :: [a] -> [[a]]
    powerset xs = filterM (\x -> [True, False]) xs