## Body

### Summary

When `cache_control_injection_points` is configured for Anthropic Claude models on a LiteLLM proxy that serves the OpenAI Agents JS SDK via `/v1/responses`, two things happen — both bad:

1. **Cache injection is a silent no-op.** The hook iterates `messages[]` looking for `role: "system"` but the Responses API protocol places the system prompt in the top-level `instructions` string, not in the messages array. The hook finds nothing, returns empty (`cache_controls=[]` in DEBUG logs), and zero `cache_creation_input_tokens`/`cache_read_input_tokens` are returned by Anthropic on any subsequent turn — caching never engages. 2. **The mere presence of the hook in the request pipeline corrupts the conversation.** Even though the injection itself does nothing, on multi-turn tool calls Claude (Opus 4.6, but reproduced across reasoning effort levels and with extended thinking disabled) repeatedly re-issues the same tool call after receiving its result. Every turn returns `200 OK` — no protocol error — but the model behaves as if the prior tool result was never consumed. The orchestrator hits its `maxTurns` limit (default 15 in `@openai/agents`) and the request errors out…