Itertools functions build iterators, so their results are produced as you consume them. groupby is especially easy to misread: it groups adjacent equal keys, and a later occurrence starts a new group. Consume each group before advancing the outer iterator. accumulate emits prefix results, while chain.from_iterable removes one collection level without creating an intermediate flattened list.
Functools handles reusable callable behavior. partial binds an argument, and lru_cache can reuse a result for repeated hashable arguments. You will inspect cache hits and misses rather than guess whether caching helped. Each exercise returns a plain materialized result for checking. Empty iterators, blank labels, negative movements, and repeated arguments are part of the contract, not exceptional cases.