feat: add header for task kind (#5142)

Add a header in the responses API request for the task kind (compact,
review, ...) for observability purpose
The header name is `codex-task-type`

codex-rs/core/src/codex.rs

@@ -99,6 +99,7 @@ use crate::rollout::RolloutRecorderParams;
 use crate::shell;
 use crate::state::ActiveTurn;
 use crate::state::SessionServices;
+use crate::state::TaskKind;
 use crate::tasks::CompactTask;
 use crate::tasks::RegularTask;
 use crate::tasks::ReviewTask;
@@ -1634,6 +1635,7 @@ pub(crate) async fn run_task(
     turn_context: Arc<TurnContext>,
     sub_id: String,
     input: Vec<InputItem>,
+    task_kind: TaskKind,
 ) -> Option<String> {
     if input.is_empty() {
         return None;
@@ -1717,6 +1719,7 @@ pub(crate) async fn run_task(
             Arc::clone(&turn_diff_tracker),
             sub_id.clone(),
             turn_input,
+            task_kind,
         )
         .await
         {
@@ -1942,6 +1945,7 @@ async fn run_turn(
     turn_diff_tracker: SharedTurnDiffTracker,
     sub_id: String,
     input: Vec<ResponseItem>,
+    task_kind: TaskKind,
 ) -> CodexResult<TurnRunResult> {
     let mcp_tools = sess.services.mcp_connection_manager.list_all_tools();
     let router = Arc::new(ToolRouter::from_config(
@@ -1971,6 +1975,7 @@ async fn run_turn(
             Arc::clone(&turn_diff_tracker),
             &sub_id,
             &prompt,
+            task_kind,
         )
         .await
         {
@@ -2044,6 +2049,7 @@ async fn try_run_turn(
     turn_diff_tracker: SharedTurnDiffTracker,
     sub_id: &str,
     prompt: &Prompt,
+    task_kind: TaskKind,
 ) -> CodexResult<TurnRunResult> {
     // call_ids that are part of this response.
     let completed_call_ids = prompt
@@ -2109,7 +2115,11 @@ async fn try_run_turn(
         summary: turn_context.client.get_reasoning_summary(),
     });
     sess.persist_rollout_items(&[rollout_item]).await;
-    let mut stream = turn_context.client.clone().stream(&prompt).await?;
+    let mut stream = turn_context
+        .client
+        .clone()
+        .stream_with_task_kind(prompt.as_ref(), task_kind)
+        .await?;
 
     let tool_runtime = ToolCallRuntime::new(
         Arc::clone(&router),