Brief Cursor Composer to build a multi-file feature without wandering off

You are working in Composer/Agent mode with multi-file edit access in this repo. Build the following feature end to end, but stay inside the scope defined below.

FEATURE
Add a "save search" button on the results page that stores the current filters under a name and lists saved searches in the sidebar for the logged-in user.

FILES YOU MAY TOUCH
app/search/results/page.tsx, app/search/results/SavedSearches.tsx (new), lib/searches/store.ts (new), app/api/searches/route.ts (new)

DATA CONTRACT BETWEEN THESE FILES
SavedSearch = { id: string, name: string, filters: FilterState, createdAt: string }. The API route returns SavedSearch[]; the sidebar component takes searches: SavedSearch[] as a prop.

OUT OF SCOPE — DO NOT TOUCH
Do not touch the existing filter-parsing logic in lib/search/parseFilters.ts, and do not modify the results table component itself.

WORKING RULES
- Before writing any code, list the exact files you will create or modify and the order you'll edit them in. If that plan needs a file outside the list above, stop and name it before touching it — don't expand scope silently.
- After each file edit, state which part of the feature description it satisfies.
- Do not install a new dependency without naming it and the reason first.
- Keep the data contract above literal — if a field name or shape needs to change to make this work, say so and wait rather than quietly adapting it on one side only.

WHEN DONE
List every file changed, one line each, and name anything from the feature description you could not finish and why.