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>.
from collections import defaultdict

S = input()

level_to_start = defaultdict(lambda: None)
level = 0
ans_len, ans_count = 0, 1

for i, char in enumerate(S):
    if char == "(":
        if level_to_start[level] == None: level_to_start[level] = i
        level += 1
    else:
        level -= 1
        level_to_start[level+1] = None

        if (start := level_to_start[level]) != None:
            length = i - start + 1

            if length > ans_len:
                ans_len = length
                ans_count = 1
            elif length == ans_len:
                ans_count += 1

print(ans_len, ans_count)