Responses Api

Follows the OpenAI Responses API spec: https://platform.openai.com/docs/api-reference/responses

Supports background mode with polling_via_cache for partial response retrieval. When background=true and polling_via_cache is enabled, returns a polling_id immediately and streams the response in the background, updating Redis cache.

# Normal request
curl -X POST http://localhost:4000/v1/responses     -H "Content-Type: application/json"     -H "Authorization: Bearer $LITELLM_MASTER_KEY"     -d '{
    "model": "gpt-4o",
    "input": "Tell me about AI"
}'

# Background request with polling
curl -X POST http://localhost:4000/v1/responses     -H "Content-Type: application/json"     -H "Authorization: Bearer $LITELLM_MASTER_KEY"     -d '{
    "model": "gpt-4o",
    "input": "Tell me about AI",
    "background": true
}'