agent: Switch to new web search provider (#29951)

Release Notes:

- N/A

crates/web_search_providers/src/cloud.rs

@@ -7,7 +7,9 @@ use gpui::{App, AppContext, Context, Entity, Subscription, Task};
 use http_client::{HttpClient, Method};
 use language_model::{LlmApiToken, RefreshLlmTokenListener};
 use web_search::{WebSearchProvider, WebSearchProviderId};
-use zed_llm_client::{WebSearchBody, WebSearchResponse};
+use zed_llm_client::{
+    CLIENT_SUPPORTS_EXA_WEB_SEARCH_PROVIDER_HEADER_NAME, WebSearchBody, WebSearchResponse,
+};
 
 pub struct CloudWebSearchProvider {
     state: Entity<State>,
@@ -84,6 +86,7 @@ async fn perform_web_search(
     let request = request_builder
         .header("Content-Type", "application/json")
         .header("Authorization", format!("Bearer {token}"))
+        .header(CLIENT_SUPPORTS_EXA_WEB_SEARCH_PROVIDER_HEADER_NAME, "true")
         .body(serde_json::to_string(&body)?.into())?;
     let mut response = http_client
         .send(request)