feat: experimental menu (#8071)

This will automatically render any `Stage::Beta` features.

The change only gets applied to the *next session*. This started as a
bug but actually this is a good thing to prevent out of distribution
push

<img width="986" height="288" alt="Screenshot 2025-12-15 at 15 38 35"
src="https://github.com/user-attachments/assets/78b7a71d-0e43-4828-a118-91c5237909c7"
/>


<img width="509" height="109" alt="Screenshot 2025-12-15 at 17 35 44"
src="https://github.com/user-attachments/assets/6933de52-9b66-4abf-b58b-a5f26d5747e2"
/>

codex-rs/codex-api/src/endpoint/responses.rs

@@ -32,6 +32,7 @@ pub struct ResponsesOptions {
     pub store_override: Option<bool>,
     pub conversation_id: Option<String>,
     pub session_source: Option<SessionSource>,
+    pub extra_headers: HeaderMap,
 }
 
 impl<T: HttpTransport, A: AuthProvider> ResponsesClient<T, A> {
@@ -73,6 +74,7 @@ impl<T: HttpTransport, A: AuthProvider> ResponsesClient<T, A> {
             store_override,
             conversation_id,
             session_source,
+            extra_headers,
         } = options;
 
         let request = ResponsesRequestBuilder::new(model, &prompt.instructions, &prompt.input)
@@ -85,6 +87,7 @@ impl<T: HttpTransport, A: AuthProvider> ResponsesClient<T, A> {
             .conversation(conversation_id)
             .session_source(session_source)
             .store_override(store_override)
+            .extra_headers(extra_headers)
             .build(self.streaming.provider())?;
 
         self.stream_request(request).await