# Minimum Canisters to Top Up an Exact Target During a layover, the ground crew must top up an auxiliary tank by **exactly** `target` units of coolant before the next leg can depart. The depot stocks coolant in sealed canisters of several fixed capacities. Each capacity is available in **unlimited** supply, and a canister must be emptied **completely** into the tank (you cannot use a partial canister). Given the target amount and the list of available canister capacities, return the **minimum number of canisters** whose capacities sum to exactly `target`. If no combination of canisters can sum to exactly `target`, return `-1`. ## Input / Output - **Input:** - `target`: a non-negative integer — the exact amount that must be added. - `capacities`: a list of distinct positive integers — the canister sizes the depot stocks. Each size may be used any number of times (including zero). - **Output:** an integer — the minimum number of canisters that sum to exactly `target`, or `-1` if it is impossible. Suggested signature: ```python def min_canisters(target: int, capacities: list[int]) -> int: ... ``` ## Constraints - `0 <= target <= 10_000` - `1 <= len(capacities) <= 100` - `1 <= capacities[i] <= 10_000` - All values in `capacities` are distinct positive integers. ## Examples **Example 1** ``` target = 11 capacities = [1, 2, 5] => 3 ``` Explanation: `5 + 5 + 1 = 11` uses 3 canisters, which is optimal. No 2-canister combination sums to 11. **Example 2** ``` target = 3 capacities = [2] => -1 ``` Explanation: Only size-2 canisters exist; no number of them sums to exactly 3. **Example 3** ``` target = 0 capacities = [1, 2, 5] => 0 ``` Explanation: Nothing needs to be added, so zero canisters are required. **Example 4** ``` target = 100 capacities = [1, 5, 10, 25] => 4 ``` Explanation: `25 * 4 = 100` uses 4 canisters, which is optimal. ## Edge Cases to Handle - `target == 0` should return `0`. - A `target` that cannot be formed by any combination must return `-1` (e.g., all capacities larger than `target`, or a parity/divisibility mismatch). - Large `target` with small capacities (the answer can be as large as `target` when a size-1 canister exists). ## Follow-up Beyond the minimum **count**, also reconstruct **one** concrete combination that achieves it: return the multiset (or sorted list) of canister capacities used. For Example 1 this would be `[1, 5, 5]`. Any combination that uses the minimum number of canisters and sums exactly to `target` is acceptable.