Beautify Houses to Strict Monotone

Given an integer array of house heights, return the minimum number of `+1` operations needed so the array is either strictly `+1`-increasing (each entry exactly one greater than the previous) or strictly `-1`-decreasing. Requirements Input: integer array nums[] (house heights). One operation increments a single entry by 1. (Decrement is not allowed in this version.) After all operations, the array must be either strictly +1-increasing (nums[i+1] = nums[i] + 1) or strictly -1-decreasing (nums[i+1] = nums[i] - 1). Return the minimum total operations across the two target shapes. Notes For the +1-increasing shape, the target row is a sequence of consecutive integers anchored at some integer a. The minimum operations is achieved by picking a = max_i (nums[i] - i); any smaller anchor would push some entry below its current height, which is impossible without decrements. The total ops is sum_i ((a + i) - nums[i]). Symmetric closed-form for the -1-decreasing shape: pick a = max_i (nums[i] + i) and total ops is sum_i ((a - i) - nums[i]). Return the minimum of the two totals. Watch for integer overflow on large arrays; the anchor a can grow with input size. Preparation Derive the closed-form by hand once; once it clicks, the implementation is two passes per shape (find a, then sum the deltas). Drill the increasing case on [1, 5, 2]: a = max(1-0, 5-1, 2-2) = 4, target [4, 5, 6], ops 3 + 0 + 4 = 7.