Add a backend API endpoint

Follow this order — each step depends on the last:

1. Schema — add or extend the Pydantic request/response models in the matching domain module under backend/app/schemas/ (e.g. prompts.py or accounts.py). Reuse existing field-validation patterns in that module (e.g. unique-name validation) rather than inventing new ones.
2. Route — add the endpoint in the appropriate router file under backend/app/api/ (routes.py for prompt endpoints, session_routes.py, account_routes.py, password_routes.py, or llm_connection_routes.py for auth, eval_routes.py for evaluations, refine_routes.py for refinement, public_routes.py for unauthenticated /api/public endpoints, admin_routes.py for admin-only; oauth_routes.py and mcp_routes.py belong to the MCP/OAuth integration). Keep the handler thin: parse/validate via the schema, call a service function, return its result. Apply the existing auth dependency pattern (backend/app/auth/dependencies.py) if the endpoint should require authentication — check how neighboring endpoints in the same file do it. Register a new router in backend/app/main.py when needed; a new account router is composed in backend/app/api/auth_routes.py instead.
3. Service logic — put any real logic in backend/app/services/ (new file or extend an existing one), not in the route handler. If the endpoint touches the database, follow the ownership-scoping pattern: services use services/prompts/ownership.py (get_owned_prompt), and route handlers use the api/deps.get_owned_prompt wrapper, which maps a missing or foreign prompt to 404; use OperationError in services and call_service in HTTP adapters to translate expected service failures at the HTTP boundary. REST and MCP share prompt services; MCP calls those services directly and maps failures to tool errors. Keep ownership checks in the shared operation. Resolve AI calls in route handlers through api/deps.get_llm_connection (not the /me/llm-connection handler or services/accounts/connections.get_llm_connection, which read the stored settings) and services/llm_client.py, with the existing budget and spend-ledger handling.
4. Tests — add tests in backend/tests/ (there's already a test_api.py / test_auth.py / test_admin_routes.py split by router — put new tests in the matching file, or create a new one if the endpoint doesn't fit any existing file). Cover: happy path, validation failure (422), auth failure if applicable, and ownership isolation if the endpoint is user-scoped. Update the module table in backend/tests/README.md when adding a test file.
5. Document the contract — update docs/api.md for public route changes. Update product/FEATURES.md — add or edit a row in the relevant table with the feature name, status, one-line description, and the file(s) it's implemented in. This file is the project's source-of-truth feature list and is easy to forget.
6. Verify — run make lint-backend && make test before considering the work done.