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/compact.rs

@@ -16,6 +16,7 @@ use crate::protocol::InputItem;
 use crate::protocol::InputMessageKind;
 use crate::protocol::TaskStartedEvent;
 use crate::protocol::TurnContextItem;
+use crate::state::TaskKind;
 use crate::truncate::truncate_middle;
 use crate::util::backoff;
 use askama::Template;
@@ -258,7 +259,11 @@ async fn drain_to_completed(
     sub_id: &str,
     prompt: &Prompt,
 ) -> CodexResult<()> {
-    let mut stream = turn_context.client.clone().stream(prompt).await?;
+    let mut stream = turn_context
+        .client
+        .clone()
+        .stream_with_task_kind(prompt, TaskKind::Compact)
+        .await?;
     loop {
         let maybe_event = stream.next().await;
         let Some(event) = maybe_event else {