### Check for existing issues

- [x] I have searched the existing issues and checked that my issue is not a duplicate.

### What happened?

When streaming Azure Responses API requests through the LiteLLM proxy, it sends `response.output_text.delta` events but **omits the required setup events** (`response.created`, `response.in_progress`, `response.output_item.added`, `response.content_part.added`), causing clients to reject the deltas.

Endpoints: `/openai/v1/responses` `/v1/responses` `/responses`

This was discovered when using an Azure model via the Lite LLM Proxy with Codex. Error: `codex_core::util: OutputTextDelta without active item`

**Expected SSE sequence (sourced from Azure OpenAI response directly):** ``` {"type":"response.created",...} {"type":"response.in_progress",...} {"type":"response.output_item.added",...} {"type":"response.content_part.added",...} {"type":"response.output_text.delta","delta":"Hello",...} {"type":"response.output_text.delta","delta":" there",...} {"type":"response.output_text.done",...} {"type":"response.content_part.done",...} {"type":"response.output_item.done",...} {"type":"response.completed",...} ``` **Ac…