Solve peak element and unique word abbreviation

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.