This is yet another problem dealing with regular bracket sequences.

We should remind you that a bracket sequence is called regular, if by inserting «+» and «1» into it we can get a correct mathematical expression. For example, sequences «(())()», «()» and «(()(()))» are regular, while «)(», «(()» and «(()))(» are not. 

You are given a string of «(» and «)» characters. You are to find its longest substring that is a regular bracket sequence. You are to find the number of such substrings as well.
The first line of the input file contains a non-empty string, consisting of «(» and «)» characters. Its length does not exceed 10<sup class="upper-index">6</sup>.
def solution():
    brackets = list(input())
    open_position_by_tally = {0: -1}  # tally count -> last_opened candidate
    tally = 0
    max_length = 0
    max_count = 0
    for idx, b in enumerate(brackets):
        if b == '(':
            tally += 1
            if tally not in open_position_by_tally:
                open_position_by_tally[tally] = idx
        else:
            tally -= 1
            if tally not in open_position_by_tally:
                open_position_by_tally[tally] = idx
            else:
                length = idx - open_position_by_tally[tally]
                if length == max_length:
                    max_count += 1
                elif length > max_length:
                    max_length = length
                    max_count = 1
            if tally + 1 in open_position_by_tally:
                del open_position_by_tally[tally + 1]

    if max_length == 0:
        print(0, 1)
    else:
        print(max_length, max_count)


solution()