fix: handle all web_search actions and in progress invocations (#9960)

### Summary
- Parse all `web_search` tool actions (`search`, `find_in_page`,
`open_page`).
- Previously we only parsed + displayed `search`, which made the TUI
appear to pause when the other actions were being used.
- Show in progress `web_search` calls as `Searching the web`
  - Previously we only showed completed tool calls

<img width="308" height="149" alt="image"
src="https://github.com/user-attachments/assets/90a4e8ff-b06a-48ff-a282-b57b31121845"
/>

### Tests
Added + updated tests, tested locally

### Follow ups
Update VSCode extension to display these as well

codex-rs/tui/src/history_cell.rs

@@ -43,6 +43,8 @@ use codex_core::protocol::FileChange;
 use codex_core::protocol::McpAuthStatus;
 use codex_core::protocol::McpInvocation;
 use codex_core::protocol::SessionConfiguredEvent;
+use codex_core::web_search::web_search_detail;
+use codex_protocol::models::WebSearchAction;
 use codex_protocol::openai_models::ReasoningEffort as ReasoningEffortConfig;
 use codex_protocol::plan_tool::PlanItemArg;
 use codex_protocol::plan_tool::StepStatus;
@@ -1342,9 +1344,89 @@ pub(crate) fn new_active_mcp_tool_call(
     McpToolCallCell::new(call_id, invocation, animations_enabled)
 }
 
-pub(crate) fn new_web_search_call(query: String) -> PrefixedWrappedHistoryCell {
-    let text: Text<'static> = Line::from(vec!["Searched".bold(), " ".into(), query.into()]).into();
-    PrefixedWrappedHistoryCell::new(text, "• ".dim(), "  ")
+fn web_search_header(completed: bool) -> &'static str {
+    if completed {
+        "Searched"
+    } else {
+        "Searching the web"
+    }
+}
+
+#[derive(Debug)]
+pub(crate) struct WebSearchCell {
+    call_id: String,
+    query: String,
+    action: Option<WebSearchAction>,
+    start_time: Instant,
+    completed: bool,
+    animations_enabled: bool,
+}
+
+impl WebSearchCell {
+    pub(crate) fn new(
+        call_id: String,
+        query: String,
+        action: Option<WebSearchAction>,
+        animations_enabled: bool,
+    ) -> Self {
+        Self {
+            call_id,
+            query,
+            action,
+            start_time: Instant::now(),
+            completed: false,
+            animations_enabled,
+        }
+    }
+
+    pub(crate) fn call_id(&self) -> &str {
+        &self.call_id
+    }
+
+    pub(crate) fn update(&mut self, action: WebSearchAction, query: String) {
+        self.action = Some(action);
+        self.query = query;
+    }
+
+    pub(crate) fn complete(&mut self) {
+        self.completed = true;
+    }
+}
+
+impl HistoryCell for WebSearchCell {
+    fn display_lines(&self, width: u16) -> Vec<Line<'static>> {
+        let bullet = if self.completed {
+            "•".dim()
+        } else {
+            spinner(Some(self.start_time), self.animations_enabled)
+        };
+        let header = web_search_header(self.completed);
+        let detail = web_search_detail(self.action.as_ref(), &self.query);
+        let text: Text<'static> = if detail.is_empty() {
+            Line::from(vec![header.bold()]).into()
+        } else {
+            Line::from(vec![header.bold(), " ".into(), detail.into()]).into()
+        };
+        PrefixedWrappedHistoryCell::new(text, vec![bullet, " ".into()], "  ").display_lines(width)
+    }
+}
+
+pub(crate) fn new_active_web_search_call(
+    call_id: String,
+    query: String,
+    animations_enabled: bool,
+) -> WebSearchCell {
+    WebSearchCell::new(call_id, query, None, animations_enabled)
+}
+
+pub(crate) fn new_web_search_call(
+    call_id: String,
+    query: String,
+    action: WebSearchAction,
+) -> WebSearchCell {
+    let mut cell = WebSearchCell::new(call_id, query, Some(action), false);
+    cell.complete();
+    cell
 }
 
 /// If the first content is an image, return a new cell with the image.
@@ -1837,6 +1919,7 @@ mod tests {
     use codex_core::config::types::McpServerConfig;
     use codex_core::config::types::McpServerTransportConfig;
     use codex_core::protocol::McpAuthStatus;
+    use codex_protocol::models::WebSearchAction;
     use codex_protocol::parse_command::ParsedCommand;
     use dirs::home_dir;
     use pretty_assertions::assert_eq;
@@ -2060,8 +2143,12 @@ mod tests {
 
     #[test]
     fn web_search_history_cell_snapshot() {
+        let query =
+            "example search query with several generic words to exercise wrapping".to_string();
         let cell = new_web_search_call(
-            "example search query with several generic words to exercise wrapping".to_string(),
+            "call-1".to_string(),
+            query.clone(),
+            WebSearchAction::Search { query: Some(query) },
         );
         let rendered = render_lines(&cell.display_lines(64)).join("\n");
 
@@ -2070,8 +2157,12 @@ mod tests {
 
     #[test]
     fn web_search_history_cell_wraps_with_indented_continuation() {
+        let query =
+            "example search query with several generic words to exercise wrapping".to_string();
         let cell = new_web_search_call(
-            "example search query with several generic words to exercise wrapping".to_string(),
+            "call-1".to_string(),
+            query.clone(),
+            WebSearchAction::Search { query: Some(query) },
         );
         let rendered = render_lines(&cell.display_lines(64));
 
@@ -2086,7 +2177,12 @@ mod tests {
 
     #[test]
     fn web_search_history_cell_short_query_does_not_wrap() {
-        let cell = new_web_search_call("short query".to_string());
+        let query = "short query".to_string();
+        let cell = new_web_search_call(
+            "call-1".to_string(),
+            query.clone(),
+            WebSearchAction::Search { query: Some(query) },
+        );
         let rendered = render_lines(&cell.display_lines(64));
 
         assert_eq!(rendered, vec!["• Searched short query".to_string()]);
@@ -2094,8 +2190,12 @@ mod tests {
 
     #[test]
     fn web_search_history_cell_transcript_snapshot() {
+        let query =
+            "example search query with several generic words to exercise wrapping".to_string();
         let cell = new_web_search_call(
-            "example search query with several generic words to exercise wrapping".to_string(),
+            "call-1".to_string(),
+            query.clone(),
+            WebSearchAction::Search { query: Some(query) },
         );
         let rendered = render_lines(&cell.transcript_lines(64)).join("\n");