feat: sync all endpoints with MITM LS bypass + real-time thinking streaming

- Responses API (streaming): MITM bypass path polls MitmStore directly
  when custom tools are active, skipping LS step polling entirely.
  Streams thinking text deltas in real-time as they arrive from the MITM.
  Handles function calls, text response, and thinking/reasoning events.

- Responses API (sync): Same MITM bypass for non-streaming responses.
  Polls MitmStore for function calls or completed text before falling
  back to LS path.

- Gemini endpoint: MITM bypass polls MitmStore directly for tool call
  responses, eliminating LS overhead.

- MitmStore: Added captured_thinking_text field with set/peek/take methods
  for real-time thinking text capture from MITM SSE.

- MITM proxy: Now captures both thinking_text and response_text from
  StreamingAccumulator into MitmStore when bypass mode is active.

.gemini/plans/sync-and-latency.md

@@ -0,0 +1,46 @@
+# Sync All Endpoints + Latency + Thinking Streaming
+
+## Phase 1: Sync Responses API (`/v1/responses`) with LS bypass
+
+Current state:
+
+- `handle_responses_stream` (line 529-859) polls LS steps for text
+- Doesn't use MitmStore bypass at all
+- Still suffers from LS multi-turn overhead when tools are active
+
+Fix:
+
+- Add MITM bypass path (same as completions) — check MitmStore for text + function calls
+- For function calls: emit `response.output_item.added` (function_call type) + done events
+- For text: stream from MitmStore `captured_response_text` + `response_complete`
+
+## Phase 2: Sync Gemini endpoint (`/v1/gemini`) with LS bypass
+
+Current state:
+
+- `handle_gemini` (line 57-236) uses `poll_for_response` then checks MitmStore
+- Already checks `take_any_function_calls()` after polling
+- But `poll_for_response` still goes through LS steps
+
+Fix:
+
+- When tools are active, poll MitmStore directly instead of `poll_for_response`
+
+## Phase 3: Latency improvements
+
+- Reduce poll intervals across all handlers
+- Add MITM store thinking_text capture for real-time streaming
+
+## Phase 4: Real-time thinking streaming investigation
+
+Current state:
+
+- Google SSE includes `thought: true` parts with thinking text
+- `streaming_acc.thinking_text` accumulates this
+- Currently only used for final usage stats, not streamed in real-time
+
+Investigation needed:
+
+- The MITM intercept already captures thinking_text per-chunk
+- Need to store thinking_text updates in MitmStore incrementally
+- Responses handler can then stream thinking deltas in real-time