feat(observability): integrate Sentry error tracking for API and Web apps

- API: add @sentry/nestjs with instrument.ts, SentryModule, and SentryGlobalFilter
- Web: add @sentry/nextjs with client/server/edge configs, instrumentation hook
- Update next.config.js with withSentryConfig wrapper
- Replace TODO in error.tsx with Sentry.captureException
- Add SENTRY_DSN, SENTRY_AUTH_TOKEN, SENTRY_ORG, SENTRY_PROJECT to .env.example

Co-Authored-By: Paperclip <noreply@paperclip.ing>

apps/api/src/app.module.ts

@@ -1,7 +1,8 @@
 import { type MiddlewareConsumer, Module, type NestModule } from '@nestjs/common';
-import { APP_GUARD } from '@nestjs/core';
+import { APP_FILTER, APP_GUARD } from '@nestjs/core';
 import { CqrsModule } from '@nestjs/cqrs';
 import { ThrottlerModule } from '@nestjs/throttler';
+import { SentryGlobalFilter, SentryModule } from '@sentry/nestjs/setup';
 import { AdminModule } from '@modules/admin';
 import { AnalyticsModule } from '@modules/analytics';
 import { AuthModule } from '@modules/auth';
@@ -20,6 +21,7 @@ import { AppController } from './app.controller';
 
 @Module({
   imports: [
+    SentryModule.forRoot(),
     CqrsModule.forRoot(),
     SharedModule,
     AuthModule,
@@ -58,6 +60,10 @@ import { AppController } from './app.controller';
   ],
   controllers: [AppController],
   providers: [
+    {
+      provide: APP_FILTER,
+      useClass: SentryGlobalFilter,
+    },
     {
       provide: APP_GUARD,
       useClass: ThrottlerBehindProxyGuard,