[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/analytics/src/client.rs

@@ -19,6 +19,7 @@ use crate::facts::SkillInvokedInput;
 use crate::facts::SubAgentThreadStartedInput;
 use crate::facts::TrackEventsContext;
 use crate::facts::TurnCodexErrorFact;
+use crate::facts::TurnProfileFact;
 use crate::facts::TurnResolvedConfigFact;
 use crate::facts::TurnTokenUsageFact;
 use crate::reducer::AnalyticsReducer;
@@ -257,6 +258,12 @@ impl AnalyticsEventsClient {
         )));
     }
 
+    pub fn track_turn_profile(&self, fact: TurnProfileFact) {
+        self.record_fact(AnalyticsFact::Custom(CustomAnalyticsFact::TurnProfile(
+            Box::new(fact),
+        )));
+    }
+
     pub fn track_turn_codex_error(&self, fact: TurnCodexErrorFact) {
         self.record_fact(AnalyticsFact::Custom(CustomAnalyticsFact::TurnCodexError(
             Box::new(fact),