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/lib/schemas.ts

@@ -219,6 +219,7 @@ export const UserSettingsSchema = z.object({
   thinkingBudget: z.enum(["low", "medium", "high"]).optional(),
   enableProLazyEditsMode: z.boolean().optional(),
   enableProSmartFilesContextMode: z.boolean().optional(),
+  enableProWebSearch: z.boolean().optional(),
   proSmartContextOption: z.enum(["balanced", "conservative"]).optional(),
   selectedTemplateId: z.string(),
   enableSupabaseWriteSqlMigration: z.boolean().optional(),