[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/facts.rs

@@ -104,6 +104,23 @@ pub struct TurnTokenUsageFact {
     pub token_usage: TokenUsage,
 }
 
+#[derive(Clone, Debug, PartialEq, Eq, Serialize)]
+pub struct TurnProfile {
+    pub before_first_sampling_ms: u64,
+    pub sampling_ms: u64,
+    pub between_sampling_overhead_ms: u64,
+    pub tool_blocking_ms: u64,
+    pub after_last_sampling_ms: u64,
+    pub sampling_request_count: u32,
+    pub sampling_retry_count: u32,
+}
+
+#[derive(Clone)]
+pub struct TurnProfileFact {
+    pub turn_id: String,
+    pub profile: TurnProfile,
+}
+
 #[derive(Clone)]
 pub struct TurnCodexErrorFact {
     pub(crate) turn_id: String,
@@ -476,6 +493,7 @@ pub(crate) enum CustomAnalyticsFact {
     GuardianReview(Box<GuardianReviewEventParams>),
     TurnResolvedConfig(Box<TurnResolvedConfigFact>),
     TurnTokenUsage(Box<TurnTokenUsageFact>),
+    TurnProfile(Box<TurnProfileFact>),
     TurnCodexError(Box<TurnCodexErrorFact>),
     SkillInvoked(SkillInvokedInput),
     AppMentioned(AppMentionedInput),