feat: show per-stage timing (듣기/LLM/TTS) in the transcript channel

The transcript channel only showed STT and LLM seconds. Add wall-clock
start/end times and durations for listening, LLM and TTS so it's obvious
what takes long; STT surfaces as the gap between listening end and LLM start.

- bridge: emit llm_start_ms/llm_end_ms on meta and tts_*_ms on the end event
- bot: capture the listening window, assemble full timing after the stream,
  and render a per-stage breakdown in the transcript message

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

bot/src/bridge.ts

@@ -50,6 +50,19 @@ export interface ConverseMeta {
   /** Per-stage timing (seconds) for diagnosing latency. */
   stt_sec?: number;
   think_sec?: number;
+  /** Wall-clock LLM window (epoch ms) so the transcript channel can show when
+   *  the reply engine started/finished. */
+  llm_start_ms?: number;
+  llm_end_ms?: number;
+}
+
+/** Final event of a streamed turn, carrying TTS timing (synthesis runs after
+ *  the meta line, so it can't be reported there). */
+export interface ConverseEnd {
+  tts_sec?: number;
+  /** Wall-clock TTS window (epoch ms). */
+  tts_start_ms?: number;
+  tts_end_ms?: number;
 }
 
 export interface ConverseStreamHandlers {
@@ -57,6 +70,8 @@ export interface ConverseStreamHandlers {
   onMeta?: (meta: ConverseMeta) => void | Promise<void>;
   /** Fired per sentence as its audio finishes synthesising (in order). */
   onAudio?: (wav: Buffer) => void | Promise<void>;
+  /** Fired once after the last clip, with TTS timing. */
+  onEnd?: (end: ConverseEnd) => void | Promise<void>;
 }
 
 /** Parse a byte stream of newline-delimited JSON into objects, one per line. */
@@ -102,6 +117,8 @@ export async function converseStream(
     } else if (ev.type === "audio" && ev.audio_b64) {
       const clip = decodeWav(ev.audio_b64);
       if (clip) await handlers.onAudio?.(clip);
+    } else if (ev.type === "end") {
+      await handlers.onEnd?.(ev as ConverseEnd);
     }
   }
 }