You are given two independent coding problems to solve. 1) Peak element in an array (binary search) - Input: an integer array `nums` of length `n >= 1`. - A peak is an index `i` such that `nums[i] > nums[i-1]` and `nums[i] > nums[i+1]`. - Treat out-of-bounds neighbors as negative infinity: `nums[-1] = nums[n] = -∞`. - Task: return the index of any peak element. - Requirement: design an algorithm with `O(log n)` time. 2) Minimal unique abbreviations for a dictionary - Input: an array of distinct strings `words`. - A word can be abbreviated as: `prefix + <number of omitted middle characters> + last_char` (e.g., `"international" -> "i11l"`). - Abbreviations must be unique across the list. - If an abbreviation is not shorter than the original word, keep the original word. - Task: return an array `abbr` where `abbr[i]` is the minimal-length abbreviation for `words[i]` that is unique among all words. Clarify any assumptions you need (e.g., lowercase letters only) and implement the solutions.