Support web search (#1370)

<!-- This is an auto-generated description by cubic. -->

## Summary by cubic
Adds web search to Dyad Pro chats with a new UI, tag parsing, and a Pro
Mode toggle that wires through to the engine.

- **New Features**
  - Pro Mode toggle: “Web Search” (settings.enableProWebSearch).
  - New custom tags: dyad-web-search, dyad-web-search-result, dyad-read.
- Collapsible Web Search Result UI with in-progress badge and markdown
rendering.
- Engine integration: passes enable_web_search and activates DyadEngine
when web search is on.

<!-- End of auto-generated description by cubic. -->

src/components/ProModeSelector.tsx

@@ -19,6 +19,12 @@ import { hasDyadProKey, type UserSettings } from "@/lib/schemas";
 export function ProModeSelector() {
   const { settings, updateSettings } = useSettings();
 
+  const toggleWebSearch = () => {
+    updateSettings({
+      enableProWebSearch: !settings?.enableProWebSearch,
+    });
+  };
+
   const toggleLazyEdits = () => {
     updateSettings({
       enableProLazyEditsMode: !settings?.enableProLazyEditsMode,
@@ -105,6 +111,15 @@ export function ProModeSelector() {
               settingEnabled={Boolean(settings?.enableDyadPro)}
               toggle={toggleProEnabled}
             />
+            <SelectorRow
+              id="web-search"
+              label="Web Search"
+              description="Search the web for information"
+              tooltip="Uses the web to search for information"
+              isTogglable={proModeTogglable}
+              settingEnabled={Boolean(settings?.enableProWebSearch)}
+              toggle={toggleWebSearch}
+            />
             <SelectorRow
               id="lazy-edits"
               label="Turbo Edits"