# Solution Alignment The prompt asks for an implementation-level answer. The safest way to present it is to define the state, maintain clear invariants, then walk through complexity and tests. ## Problem Restatement Implement a function that, given an integer array nums and an integer target T, returns all unique triplets [x, y, z] from nums such that x + y + z = T. Requirements: avoid duplicate triplets (enforce x <= y <= z within each triplet), return results in lexicographic order, and aim for O(n^ 2) time using sorting and a two-pointer strategy with O( 1) extra space beyond the output. Discuss edge cases (duplicates, all zeros, no solution, mixed signs, large n) and provide a concise set of test cases. Analyze time and space complexity. ## Recommended Approach Model the states explicitly and use BFS for unweighted shortest paths, Dijkstra for weighted non-negative paths, or topological DP for DAGs. Track visited states at the right granularity so cycles do not cause repeated work. ## Correctness The implementation should maintain an invariant after each loop or operation that directly matches the problem statement. At termination, that invariant implies the returned value has considered every valid candidate exactly once, or has preserved the required data-structure state after every API call. ## Complexity BFS is O(V + E) time and O(V) space for a standard graph. Expanded-state problems multiply those bounds by the number of state dimensions. ## Edge Cases and Tests Disconnected graph, source equals target, cycles, duplicate edges, unreachable target, and whether the answer counts nodes, edges, moves, or transfers.