feat(responses): implement full multi-turn support (#2295)

I think the implementation needs more simplification. Spent way too much
time trying to get the tests pass with models not co-operating :(
Finally had to switch claude-sonnet to get things to pass reliably.

### Test Plan

```
export TAVILY_SEARCH_API_KEY=...
export OPENAI_API_KEY=...

uv run pytest -p no:warnings \
   -s -v tests/verifications/openai_api/test_responses.py \
 --provider=stack:starter \
  --model openai/gpt-4o
```

tests/verifications/openai_api/fixtures/test_cases/responses.yaml

@@ -36,7 +36,7 @@ test_response_mcp_tool:
   test_params:
     case:
     - case_id: "boiling_point_tool"
-      input: "What is the boiling point of polyjuice?"
+      input: "What is the boiling point of myawesomeliquid in Celsius?"
       tools:
       - type: mcp
         server_label: "localmcp"
@@ -94,3 +94,43 @@ test_response_multi_turn_image:
         output: "llama"
       - input: "What country do you find this animal primarily in? What continent?"
         output: "peru"
+
+test_response_multi_turn_tool_execution:
+  test_name: test_response_multi_turn_tool_execution
+  test_params:
+    case:
+    - case_id: "user_file_access_check"
+      input: "I need to check if user 'alice' can access the file 'document.txt'. First, get alice's user ID, then check if that user ID can access the file 'document.txt'. Do this as a series of steps, where each step is a separate message. Return only one tool call per step. Summarize the final result with a single 'yes' or 'no' response."
+      tools:
+      - type: mcp
+        server_label: "localmcp"
+        server_url: "<FILLED_BY_TEST_RUNNER>"
+      output: "yes"
+    - case_id: "experiment_results_lookup"
+      input: "I need to get the results for the 'boiling_point' experiment. First, get the experiment ID for 'boiling_point', then use that ID to get the experiment results. Tell me what you found."
+      tools:
+      - type: mcp
+        server_label: "localmcp"
+        server_url: "<FILLED_BY_TEST_RUNNER>"
+      output: "100°C"
+
+test_response_multi_turn_tool_execution_streaming:
+  test_name: test_response_multi_turn_tool_execution_streaming
+  test_params:
+    case:
+    - case_id: "user_permissions_workflow"
+      input: "Help me with this security check: First, get the user ID for 'charlie', then get the permissions for that user ID, and finally check if that user can access 'secret_file.txt'. Stream your progress as you work through each step."
+      tools:
+      - type: mcp
+        server_label: "localmcp"
+        server_url: "<FILLED_BY_TEST_RUNNER>"
+      stream: true
+      output: "no"
+    - case_id: "experiment_analysis_streaming"
+      input: "I need a complete analysis: First, get the experiment ID for 'chemical_reaction', then get the results for that experiment, and tell me if the yield was above 80%. Please stream your analysis process."
+      tools:
+      - type: mcp
+        server_label: "localmcp"
+        server_url: "<FILLED_BY_TEST_RUNNER>"
+      stream: true
+      output: "85%"