# 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 1) Given a string itemCategories (lowercase English letters) and an integer k (0 ≤ k ≤ 26), count the number of indices i (1 ≤ i < n) that partition the string into a non-empty prefix and a non-empty suffix such that the number of distinct characters that appear in both the prefix and the suffix is strictly greater than k. Return the count. Constraints: 1 ≤ n ≤ 1e5. 2) Given an array itemCosts of n item prices, create packages where each package contains at most two items and every package has the same total cost S. Each item can be used in at most one package. Choose S to maximize the number of packages and return that maximum. Single-item packages are allowed only if their cost equals S; t... ## Recommended Approach Use the string constraints to choose between two pointers, a stack, frequency counts, prefix/suffix state, or dynamic programming. Maintain the invariant that processed characters have already been normalized, counted, or matched according to the operation. ## 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 Most direct string scans are O(n) time. Space ranges from O(1) for two pointers to O(n) for stacks, maps, or DP tables. ## Edge Cases and Tests Empty string, length 1, repeated characters, invalid characters, case sensitivity, Unicode vs ASCII, and very long input.