feat: add trace-level input/output to Langfuse observability (#69)

* feat: add trace-level input/output to Langfuse observability

- Add @langfuse/client and @langfuse/tracing dependencies
- Wrap POST handler with observe() for proper tracing
- Use updateActiveTrace() to set trace input, output, sessionId, userId
- Filter Next.js HTTP spans in shouldExportSpan so AI SDK spans become root traces
- Enable recordInputs/recordOutputs in experimental_telemetry

* refactor: extract Langfuse logic to separate lib/langfuse.ts module

instrumentation.ts

@@ -12,11 +12,24 @@ export function register() {
     publicKey: process.env.LANGFUSE_PUBLIC_KEY,
     secretKey: process.env.LANGFUSE_SECRET_KEY,
     baseUrl: process.env.LANGFUSE_BASEURL,
+    // Filter out Next.js HTTP request spans so AI SDK spans become root traces
+    shouldExportSpan: ({ otelSpan }) => {
+      const spanName = otelSpan.name;
+      // Skip Next.js HTTP infrastructure spans
+      if (spanName.startsWith('POST /') ||
+          spanName.startsWith('GET /') ||
+          spanName.includes('BaseServer') ||
+          spanName.includes('handleRequest')) {
+        return false;
+      }
+      return true;
+    },
   });
 
   const tracerProvider = new NodeTracerProvider({
     spanProcessors: [langfuseSpanProcessor],
   });
 
+  // Register globally so AI SDK's telemetry also uses this processor
   tracerProvider.register();
 }