Responses API docs and JSON examples

[enyst]

This PR adds documentation and concrete JSON examples for the OpenAI Responses API when used natively via LiteLLM (e.g., gpt-5).

What’s included

• docs/responses-api.md — Overview of OpenAI Responses API via LiteLLM (gpt-5), stateless/stateful, tools, reasoning (plain + encrypted), and MCP notes.
• examples/responses_api/* — 8 JSON examples:
  1. 01_system_user_images.json
  2. 02_tool_call_and_output.json
  3. 02b_tool_call_followup_with_output_item.json
  4. 03_stateless_reasoning_plain.json
  5. 03b_stateless_reasoning_encrypted.json
  6. 04_stateful_reasoning_plain.json
  7. 04b_stateful_reasoning_encrypted.json
  8. 05_reasoning_response_examples.json

Notable correctness details

• FunctionCallOutput.output must be a JSON string in the OpenAI SDK (1.108.1). The examples and docs reflect this.
• Encrypted reasoning requires include=["reasoning.encrypted_content"].
• For stateful flows with LiteLLM, reuse the encoded previous_response_id; LiteLLM decodes upstream.

Follow-ups

• Happy to undraft when approved or make further tweaks to examples/docs if desired.

Co-authored-by: openhands openhands@all-hands.dev

@enyst can click here to continue refining the PR

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✨ Finishing touches

🧪 Generate unit tests

• Create PR with unit tests
• Post copyable unit tests in a comment
• Commit unit tests in branch docs/responses-api-clean

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