Phase 3 scaffold: chat engine (WebSocket, FCM, pricing, timer, extension, history)

- Backend: WebSocket plugin, chat/pricing/timer/extension/closure/notification services
- Client app: ChatBloc, pricing dialog, chat screen with message status, extension/goodbye flow, history
- Mitra app: MitraChatBloc, ExtensionBloc, chat screen, extension accept/reject, history
- Control center: free trial, extension timeout, early end config toggles
- DB migration: chat_messages, session_closures, session_extensions, customer_transactions tables

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

backend/package-lock.json

@@ -10,6 +10,7 @@
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         "@fastify/cors": "^9.0.1",
         "@fastify/sensible": "^5.6.0",
+        "@fastify/websocket": "^11.2.0",
         "dotenv": "^16.4.5",
         "fastify": "^4.28.1",
         "firebase-admin": "^12.2.0",
@@ -88,6 +89,43 @@
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       }
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+      "resolved": "https://registry.npmjs.org/@fastify/websocket/-/websocket-11.2.0.tgz",
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+      "funding": [
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+          "url": "https://github.com/sponsors/fastify"
+        },
+        {
+          "type": "opencollective",
+          "url": "https://opencollective.com/fastify"
+        }
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+      "license": "MIT",
+      "dependencies": {
+        "duplexify": "^4.1.3",
+        "fastify-plugin": "^5.0.0",
+        "ws": "^8.16.0"
+      }
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+      "funding": [
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+          "type": "github",
+          "url": "https://github.com/sponsors/fastify"
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+          "type": "opencollective",
+          "url": "https://opencollective.com/fastify"
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+      "license": "MIT"
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-      "optional": true,
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-      "license": "MIT",
-      "optional": true
+      "license": "MIT"
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-      "optional": true,
       "dependencies": {
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-      "license": "MIT",
-      "optional": true
+      "license": "MIT"
     },
     "node_modules/uuid": {
       "version": "10.0.0",
@@ -2737,8 +2768,28 @@
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       "resolved": "https://registry.npmjs.org/wrappy/-/wrappy-1.0.2.tgz",
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-      "license": "ISC",
-      "optional": true
+      "license": "ISC"
+    },
+    "node_modules/ws": {
+      "version": "8.20.0",
+      "resolved": "https://registry.npmjs.org/ws/-/ws-8.20.0.tgz",
+      "integrity": "sha512-sAt8BhgNbzCtgGbt2OxmpuryO63ZoDk/sqaB/znQm94T4fCEsy/yV+7CdC1kJhOU9lboAEU7R3kquuycDoibVA==",
+      "license": "MIT",
+      "engines": {
+        "node": ">=10.0.0"
+      },
+      "peerDependencies": {
+        "bufferutil": "^4.0.1",
+        "utf-8-validate": ">=5.0.2"
+      },
+      "peerDependenciesMeta": {
+        "bufferutil": {
+          "optional": true
+        },
+        "utf-8-validate": {
+          "optional": true
+        }
+      }
     },
     "node_modules/xtend": {
       "version": "4.0.2",