Public page SEO and production analytics (#125)

## Summary
- Closes the public-page gaps from #124: each route gets its own title, description, canonical, heading, links, and JSON-LD, and unknown URLs have a real 404 document.
- Adds HeyCatch and the Google tag (`G-V12DLRLPWR`) only on the production site and the iOS and Android apps. Tianji still loads on beta.
- Needs the server-infra change that stops serving the app shell for missing paths.

## Test plan
- [ ] Build production and confirm `/` and `/pricing/` HTML include a unique title, description, canonical, one h1, social tags, related links, and Organization JSON-LD.
- [ ] Confirm a nonsense path is served as `404.html` once the nginx change is deployed.
- [ ] Confirm beta and local web do not load HeyCatch or gtag, and that a production or native build does.
- [ ] Open the live site after deploy and click around, then mark HeyCatch installed in its dashboard.

Reviewed-on: #125

llm-fe/package-lock.json

@@ -19,6 +19,7 @@
         "@emotion/cache": "^11.14.0",
         "@emotion/react": "^11.14.0",
         "@emotion/styled": "^11.14.0",
+        "@heycatch/sdk": "^0.8.3",
         "@mui/icons-material": "^5.16.11",
         "@mui/material": "^5.16.11",
         "@testing-library/jest-dom": "^6.6.3",
@@ -3083,6 +3084,103 @@
         "node": "^12.22.0 || ^14.17.0 || >=16.0.0"
       }
     },
+    "node_modules/@heycatch/sdk": {
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+      "resolved": "https://registry.npmjs.org/@heycatch/sdk/-/sdk-0.8.3.tgz",
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+      "license": "MIT",
+      "dependencies": {
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+      },
+      "engines": {
+        "node": ">=22"
+      },
+      "peerDependencies": {
+        "expo-router": "*",
+        "react": "^18.2.0 || ^19.0.0"
+      },
+      "peerDependenciesMeta": {
+        "expo-router": {
+          "optional": true
+        },
+        "react": {
+          "optional": true
+        }
+      }
+    },
+    "node_modules/@heycatch/sdk/node_modules/posthog-react-native": {
+      "version": "4.81.2",
+      "resolved": "https://registry.npmjs.org/posthog-react-native/-/posthog-react-native-4.81.2.tgz",
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+        "@posthog/types": "^1.416.0"
+      },
+      "bin": {
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+      },
+      "peerDependencies": {
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+        "@react-native-async-storage/async-storage": ">=1.0.0",
+        "@react-navigation/native": ">= 5.0.0",
+        "expo-application": ">= 4.0.0",
+        "expo-device": ">= 4.0.0",
+        "expo-file-system": ">= 13.0.0",
+        "expo-localization": ">= 11.0.0",
+        "expo-updates": ">= 0.25.0",
+        "posthog-react-native-session-replay": ">= 1.6.0",
+        "react-native-device-info": ">= 10.0.0",
+        "react-native-localize": ">= 3.0.0",
+        "react-native-navigation": ">= 6.0.0",
+        "react-native-safe-area-context": ">= 4.0.0",
+        "react-native-svg": ">= 15.0.0"
+      },
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+        "@react-native-async-storage/async-storage": {
+          "optional": true
+        },
+        "@react-navigation/native": {
+          "optional": true
+        },
+        "expo-application": {
+          "optional": true
+        },
+        "expo-device": {
+          "optional": true
+        },
+        "expo-file-system": {
+          "optional": true
+        },
+        "expo-localization": {
+          "optional": true
+        },
+        "expo-updates": {
+          "optional": true
+        },
+        "posthog-react-native-session-replay": {
+          "optional": true
+        },
+        "react-native-device-info": {
+          "optional": true
+        },
+        "react-native-localize": {
+          "optional": true
+        },
+        "react-native-navigation": {
+          "optional": true
+        },
+        "react-native-safe-area-context": {
+          "optional": true
+        },
+        "react-native-svg": {
+          "optional": true
+        }
+      }
+    },
     "node_modules/@humanwhocodes/config-array": {
       "version": "0.13.0",
       "resolved": "https://registry.npmjs.org/@humanwhocodes/config-array/-/config-array-0.13.0.tgz",
@@ -5155,6 +5253,21 @@
         "url": "https://opencollective.com/popperjs"
       }
     },
+    "node_modules/@posthog/core": {
+      "version": "1.59.1",
+      "resolved": "https://registry.npmjs.org/@posthog/core/-/core-1.59.1.tgz",
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+      "license": "MIT",
+      "dependencies": {
+        "@posthog/types": "^1.416.0"
+      }
+    },
+    "node_modules/@posthog/types": {
+      "version": "1.416.0",
+      "resolved": "https://registry.npmjs.org/@posthog/types/-/types-1.416.0.tgz",
+      "integrity": "sha512-0/9+02s/0uDyrH4G/qUg1yyl6gXUg8Uuxw/PlDad3KLzo9FFS9zS/kAQaEshx/saVGupj1aAy/4EUBxWisBogw==",
+      "license": "MIT"
+    },
     "node_modules/@prettier/plugin-xml": {
       "version": "2.2.0",
       "resolved": "https://registry.npmjs.org/@prettier/plugin-xml/-/plugin-xml-2.2.0.tgz",