[Feature]: python doctor.py should ping the LLM with a real prompt

[Aditya-00a]

What problem does this solve?

agent/doctor.py already checks "is Ollama running" and "is the configured model pulled". But it doesn't actually send a test prompt — so a half-broken Ollama install (model corrupted, GPU OOM on first inference, etc.) won't get caught until the loop crashes mid-cycle.

What to do

Add an 11th check, check_llm_inference():

1. After the model-pulled check passes, send a minimal prompt to the LLM
2. Use a 5-second timeout
3. Prompt: "Reply with the single word 'ok' and nothing else."
4. Pass if the response contains "ok" (case-insensitive); fail otherwise with the response in detail

Proposed solution

Acceptance

• python doctor.py shows an 11th line for "LLM inference works"
• --json mode includes the new check
• A broken model (e.g. corrupted weights) makes the check fail with a useful error message

Notes

• Use the existing OpenAI client setup (backend/services/llm_client.py) so you get the same code path the loop uses
• Don't add a new Ollama-specific HTTP call; go through the abstraction

Alternatives you considered

No response

Are you up for contributing this?

• Yes, I'd like to send a PR
• Yes, with some guidance
• No, just suggesting