### Check for existing issues

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

### What happened?

When using LiteLLM as an Anthropic-compatible proxy for Claude Code, `/v1/messages` streaming can drop the leading text immediately after a `thinking` block finishes and the final `text` block begins.

This is not just a Claude Code rendering issue. The missing prefix is already visible in the raw LiteLLM SSE stream.

Tested paths:

1. `Claude Code -> LiteLLM /v1/messages -> anthropic provider -> vLLM` 2. `Claude Code -> LiteLLM /v1/messages -> hosted_vllm provider -> vLLM`

The issue reproduces across multiple backend models, so this does not appear to be provider-specific or model-specific. It appears to be in the common Anthropic `/v1/messages` streaming adapter/output path.

### Expected behavior

After an extended thinking block, LiteLLM should preserve the first generated `text_delta` when it starts the final `text` content block.

For example, if the prompt requires the final answer to start with:

```text FINAL: ABC ```

then the first `content_block_delta` for the final text block should include that prefix, for example:

```text event: co…