Inverse-Depth Weighted Sum of a Nested Integer List

You are given a nested list of integers, written as a string in bracket notation such as `"[3,[2,[5]]]"`. Each element of a list is either an integer or another list. The **depth** of an integer is the number of lists that contain it, counting the outermost list. In `"[3,[2,[5]]]"`, `3` has depth 1, `2` has depth 2 and `5` has depth 3. Let `maxDepth` be the largest depth of any integer in the input. The **weight** of an integer is `maxDepth - depth + 1`, so the most deeply nested integers have weight 1 and the integers directly inside the outermost list have the largest weight. Return the sum, over every integer in the input, of the integer multiplied by its weight. ### Function Signature ```python def inverse_depth_weighted_sum(nested: str) -> int: ``` ### Rules - The string follows this grammar and contains no whitespace: a list is `[`, then zero or more elements separated by `,`, then `]`; an element is an integer or a list; an integer is an optional `-` followed by decimal digits, with no leading zeros other than the single digit `0` (so `-0` never appears). - The whole string is exactly one list, the outermost one. - Only integers determine `maxDepth`. Empty lists, however deeply nested, contribute nothing and do not raise `maxDepth`. - If the input contains no integers at all, for example `"[]"` or `"[[],[[]]]"`, return `0`. ### Constraints - `2 <= len(nested) <= 10^5` - `nested` is valid under the grammar above. - Every integer is between `-100` and `100` inclusive. - Lists are nested at most 50 deep, counting the outermost list. - Under these bounds the result fits in a signed 32-bit integer. ### Examples **Example 1** - Input: `nested = "[[2,2],3,[2,2]]"` - Output: `14` - Explanation: `maxDepth` is 2. The four `2`s have depth 2 and weight 1, contributing `8`; the `3` has depth 1 and weight 2, contributing `6`. **Example 2** - Input: `nested = "[3,[2,[5]]]"` - Output: `18` - Explanation: `maxDepth` is 3, so the weights of `3`, `2` and `5` are 3, 2 and 1: `3*3 + 2*2 + 5*1 = 18`. **Example 3** - Input: `nested = "[2,[[]],[-1]]"` - Output: `3` - Explanation: The empty list inside `[[]]` holds no integer, so `maxDepth` is 2, not 3. The result is `2*2 + (-1)*1 = 3`.