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/chat/DyadMarkdownParser.tsx

@@ -19,6 +19,9 @@ import { DyadProblemSummary } from "./DyadProblemSummary";
 import { IpcClient } from "@/ipc/ipc_client";
 import { DyadMcpToolCall } from "./DyadMcpToolCall";
 import { DyadMcpToolResult } from "./DyadMcpToolResult";
+import { DyadWebSearchResult } from "./DyadWebSearchResult";
+import { DyadWebSearch } from "./DyadWebSearch";
+import { DyadRead } from "./DyadRead";
 
 interface DyadMarkdownParserProps {
   content: string;
@@ -124,6 +127,9 @@ function preprocessUnclosedTags(content: string): {
     "dyad-chat-summary",
     "dyad-edit",
     "dyad-codebase-context",
+    "dyad-web-search-result",
+    "dyad-web-search",
+    "dyad-read",
     "think",
     "dyad-command",
     "dyad-mcp-tool-call",
@@ -193,6 +199,9 @@ function parseCustomTags(content: string): ContentPiece[] {
     "dyad-chat-summary",
     "dyad-edit",
     "dyad-codebase-context",
+    "dyad-web-search-result",
+    "dyad-web-search",
+    "dyad-read",
     "think",
     "dyad-command",
     "dyad-mcp-tool-call",
@@ -282,6 +291,40 @@ function renderCustomTag(
   const { tag, attributes, content, inProgress } = tagInfo;
 
   switch (tag) {
+    case "dyad-read":
+      return (
+        <DyadRead
+          node={{
+            properties: {
+              path: attributes.path || "",
+            },
+          }}
+        >
+          {content}
+        </DyadRead>
+      );
+    case "dyad-web-search":
+      return (
+        <DyadWebSearch
+          node={{
+            properties: {},
+          }}
+        >
+          {content}
+        </DyadWebSearch>
+      );
+    case "dyad-web-search-result":
+      return (
+        <DyadWebSearchResult
+          node={{
+            properties: {
+              state: getState({ isStreaming, inProgress }),
+            },
+          }}
+        >
+          {content}
+        </DyadWebSearchResult>
+      );
     case "think":
       return (
         <DyadThink