fix(core): Remove json output schema form the next speaker check prompt (#5325)

Co-authored-by: gemini-code-assist[bot] <176961590+gemini-code-assist[bot]@users.noreply.github.com>

packages/core/src/telemetry/clearcut-logger/clearcut-logger.ts

@@ -20,6 +20,7 @@ import {
   LoopDetectedEvent,
   NextSpeakerCheckEvent,
   SlashCommandEvent,
+  MalformedJsonResponseEvent,
 } from '../types.js';
 import { EventMetadataKey } from './event-metadata-key.js';
 import { Config } from '../../config/config.js';
@@ -42,6 +43,7 @@ const flash_fallback_event_name = 'flash_fallback';
 const loop_detected_event_name = 'loop_detected';
 const next_speaker_check_event_name = 'next_speaker_check';
 const slash_command_event_name = 'slash_command';
+const malformed_json_response_event_name = 'malformed_json_response';
 
 export interface LogResponse {
   nextRequestWaitMs?: number;
@@ -557,6 +559,21 @@ export class ClearcutLogger {
     this.flushIfNeeded();
   }
 
+  logMalformedJsonResponseEvent(event: MalformedJsonResponseEvent): void {
+    const data = [
+      {
+        gemini_cli_key:
+          EventMetadataKey.GEMINI_CLI_MALFORMED_JSON_RESPONSE_MODEL,
+        value: JSON.stringify(event.model),
+      },
+    ];
+
+    this.enqueueLogEvent(
+      this.createLogEvent(malformed_json_response_event_name, data),
+    );
+    this.flushIfNeeded();
+  }
+
   logEndSessionEvent(event: EndSessionEvent): void {
     const data = [
       {