[codex] Add turn profiling analytics (#26484)

## Summary

Add flat profiling fields to `codex_turn_event` so analytics can explain
where turn wall-clock time is spent without changing tool execution
behavior.

The profile reports:
- time before the first sampling request
- sampling time across all attempts and follow-ups
- overhead between sampling requests
- time blocked in the post-sampling tool drain
- time after the final sampling request
- sampling request and retry counts

## Implementation

- Extend the existing turn timing state with constant-memory phase
accounting and one RAII phase guard.
- Observe sampling and the existing post-sampling drain only at turn
orchestration boundaries.
- Keep tool runtime, tool futures, response item handling, and turn
lifecycle values unchanged.
- Add the profiling fields directly to the existing analytics turn event
without changing app-server protocol or rollout persistence.
- Use the existing turn `status` to distinguish completed, failed, and
interrupted profiles.

Exact sampling/tool overlap is intentionally omitted because measuring
tool completion accurately would require hooks in the tool execution
path.

## Validation

- Add app-server end-to-end coverage for a single-sampling turn with no
blocking tool work.
- Add app-server end-to-end coverage for `request_user_input` blocking
followed by a second sampling request.
- CI is running on the PR; tests were not executed locally per
repository guidance.

codex-rs/core/src/tasks/mod.rs

@@ -33,6 +33,7 @@ use crate::session::turn_context::TurnContext;
 use crate::state::ActiveTurn;
 use crate::state::RunningTask;
 use crate::state::TaskKind;
+use codex_analytics::TurnProfileFact;
 use codex_analytics::TurnTokenUsageFact;
 use codex_login::AuthManager;
 use codex_models_manager::manager::SharedModelsManager;
@@ -751,6 +752,12 @@ impl Session {
             .turn_timing_state
             .time_to_first_token_ms()
             .await;
+        self.services
+            .analytics_events_client
+            .track_turn_profile(TurnProfileFact {
+                turn_id: turn_context.sub_id.clone(),
+                profile: turn_context.turn_timing_state.complete_profile(),
+            });
         self.emit_turn_stop_lifecycle(turn_context.extension_data.as_ref())
             .await;
         if let Err(err) = self
@@ -868,6 +875,12 @@ impl Session {
             .turn_timing_state
             .completed_at_and_duration_ms()
             .await;
+        self.services
+            .analytics_events_client
+            .track_turn_profile(TurnProfileFact {
+                turn_id: task.turn_context.sub_id.clone(),
+                profile: task.turn_context.turn_timing_state.complete_profile(),
+            });
         let event = EventMsg::TurnAborted(TurnAbortedEvent {
             turn_id: Some(task.turn_context.sub_id.clone()),
             reason,