Enable OpenAI's hosted web_search tool on a Responses API call, restrict it by domain and location, and render the required inline citations

Documented steps

1. POST https://api.openai.com/v1/responses with 'Authorization: Bearer <API_KEY>' and 'Content-Type: application/json'.
2. Set 'model' to a supported model. Per the current docs the Responses API supports gpt-5.6, gpt-5.5, and gpt-4.1; gpt-5-search-api is the Chat Completions counterpart.
3. Add {"type":"web_search"} to the 'tools' array. The legacy 'web_search_preview' type still works for existing integrations but lacks the newer controls.
4. Optionally restrict sources with filters:{allowed_domains:[...]} or blocked_domains, up to 100 domains. Omit the http/https prefix: use 'openai.com', not 'https://openai.com/'. Subdomains are included.
5. Optionally bias results with user_location:{country:'US' (two-letter ISO), city:'...', region:'...', timezone:'<IANA zone>'}.
6. Send the request. The model decides whether to search; use tool_choice to force or forbid the tool.
7. Read the 'web_search_call' item in the output array for the search id and action type (search, open_page, or find_in_page).
8. Read the message item's 'annotations' array for entries of type 'url_citation', each carrying url, title, and character start/end indices into the text.
9. Render every url_citation as a clearly visible, clickable link. The docs state that 'inline citations must be made clearly visible and clickable in your user interface.'
10. Budget cost as a per-call tool charge PLUS standard model token rates on the search content the model consumes.