Big and small

You are given a list of integers. Your task is to form as many pairs of integers as possible so that the larger integer of a pair is at least twice the smaller integer of the pair. Each integer can belong to at most one pair.

To be precise, we say that a pair $(a,b)$ is valid if $2a \le b$.

For example, given the list $[4,5,1,4,7,8]$, at most two such pairs can be formed. One possible solution is to choose the pairs $(1,5)$ and $(4,8)$.

In a file bigsmall.py, implement a function count_pairs that is given a list of positive integers as a parameter and returns the maximal number of valid pairs.

You should implement the function so that it runs efficiently even for long lists. The function should finish immediately in the last test case of the code template too.

def count_pairs(numbers):
    # TODO

if __name__ == "__main__":
    print(count_pairs([1])) # 0
    print(count_pairs([1, 2, 3])) # 1
    print(count_pairs([1, 2, 3, 4])) # 2
    print(count_pairs([1, 1, 1, 1])) # 0
    print(count_pairs([10**9, 1, 1, 1])) # 1
    print(count_pairs([4, 5, 1, 4, 7, 8])) # 2
    print(count_pairs([1, 2, 3, 2, 4, 6])) # 3

    numbers = [(x * 999983) % 10**6 + 1 for x in range(10**5)]
    print(count_pairs(numbers)) # 41176