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Two-step check
Compute g = gcd(len(str1), len(str2)) using Euclidean algorithm.
Extract candidate = str1[0:g] (first g characters).
Check if repeating candidate gives both str1 and str2. If yes, return candidate. If no, return empty string.
The check can be done by verifying str1 + str2 == str2 + str1. If that holds, the candidate works.
