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 Ruby (2.7.1) (judge verdict: Time Limit Exceeded)

if !ENV['RUBY_THREAD_VM_STACK_SIZE']
  exec({'RUBY_THREAD_VM_STACK_SIZE'=>'1000000000'}, '/usr/bin/ruby', $0)
  return
end

def main(str)
  lines = str.split("\n")
  _, k = lines[0].split(" ").map(&:to_i)
  as = lines[1].split(" ").map(&:to_i).sort_by { |x| - x.abs }
  sub(as, k, 1, 0, 0, [], []) % (10 ** 9 + 7)
end

def sub(as, k, cur, x, num, maxs, mins)
  return cur if num == k
  return nil if x >= as.count
  return nil if as.count - x < k - num
  update = false
  if !maxs[x] || (maxs[x] < cur)
    update  = true
    maxs[x] = cur
  end
  if !mins[x] || (mins[x] > cur)
    update = true
    mins[x] = cur
  end
  return nil unless update
  x0 = sub(as, k, cur,         x + 1, num,     maxs, mins)
  x1 = sub(as, k, cur * as[x], x + 1, num + 1, maxs, mins)
  [x0, x1].compact.max
end

unless ENV["TEST"]
  puts(main($stdin.read.chomp))
  exit(0)
end

require 'minitest/autorun'

class TestA < Minitest::Test
  def test_main
    assert_equal(12, main("4 2
1 2 -3 -4"))
    assert_equal(1000000001, main("4 3
-1 -2 -3 -4"))
    assert_equal(1000000000, main("2 1
-1 1000000000"))
    assert_equal(999983200, main("10 10
1000000000 100000000 10000000 1000000 100000 10000 1000 100 10 1"))
    assert_equal(0, main("#{2 * 10 ** 5} #{2 * 10 ** 5}
#{ Array.new(2 * 10 ** 5) { 0 } }"))
  end
end