Find All Distinct Four-Value Target Groups

### Problem Given the integer array `values` and integer `goal`, return all distinct groups of four values whose sum equals `goal`, using four different positions from the array for each group. Different position selections that yield the same four values describe one result. A repeated value is allowed within a group when the array contains enough occurrences. Return the groups in any order; no canonical arrangement of a group's values is required. ### Examples **Example 1** ```text Input: values = [-3, -1, 0, 2, 4, 5], goal = 5 Output: [[-3, -1, 4, 5], [-1, 0, 2, 4]] ``` Both groups sum to five, and no other choice produces a distinct qualifying group. **Example 2** ```text Input: values = [-6, -6, 3, 3, 3, 3], goal = -6 Output: [[-6, -6, 3, 3]] ``` Multiple selections of the positions holding `3` produce the same value group, so it appears once. ### Constraints & Assumptions - The array has between 1 and 200 elements. - Every array value is from -1,000,000,000 through 1,000,000,000. - The target `goal` has those same inclusive bounds. - All four positions used for one result must differ. - Return an empty list if there are no qualifying groups, including when fewer than four positions exist. - No output ordering or particular algorithm is prescribed. ### Clarifying Questions to Ask - Should repeated index selections appear separately? No; results are unique groups of values. - Can a group use four equal values? Yes, but only if at least four distinct positions supply them and their sum matches the target. ```hint Account for intermediate arithmetic A sum of four permitted input values can exceed the range of a signed 32-bit integer, even though the target itself fits in that range. ``` ### What a Strong Answer Covers - A search that enumerates every qualifying four-value group and never reuses a position within one group. - Duplicate handling at every selection level while retaining necessary repeated values. - Safe arithmetic for four-value sums and any pruning bounds. - A proof of completeness and explicit time, working-space, and output-space costs. ### Follow-up Questions - How can minimum and maximum achievable sums prune a sorting-based search without losing answers? - Which intermediate expressions need widening in a language with fixed-width integer arithmetic?