feat: forward X-HF-Bill-To header to upstream for org billing

fix: default stream to false and drop dangling tool_choice (auto-mode classifier)

feat: support Claude Code auto-mode classifier and non-streaming requests

fix: accept system role in messages array

fix: support adaptive thinking type in request deserialization

Fix streaming tool calls and input parsing for Responses API

Support more tool types in Responses API (web_search, code_interpreter, etc.)

Add OpenAI Responses API support (/v1/responses endpoint)

Initial Rust port of anthropic-messages-proxy