fix: add ffmpeg to API container for TTS audio conversion

The Gemini TTS service uses pydub which requires ffmpeg to convert
audio formats. Previously only the Worker container had ffmpeg.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

backend/Dockerfile

@@ -3,7 +3,7 @@
 # =============================================================================
 # Stage 1: Builder - Install dependencies
 # Stage 2: Base - Common runtime for API and Worker
-# Stage 3: API - FastAPI + Gunicorn (no ffmpeg)
+# Stage 3: API - FastAPI + Gunicorn (with ffmpeg for TTS audio conversion)
 # Stage 4: Worker - Celery worker (with ffmpeg for video processing)
 # =============================================================================
 
@@ -75,6 +75,18 @@ USER app
 # -----------------------------------------------------------------------------
 FROM base AS api
 
+# Switch to root to install ffmpeg
+USER root
+
+# Install ffmpeg for TTS audio conversion
+RUN apt-get update && apt-get install -y --no-install-recommends \
+    ffmpeg \
+    && rm -rf /var/lib/apt/lists/* \
+    && apt-get clean
+
+# Switch back to non-root user
+USER app
+
 # Set production environment variables
 ENV APP_ENV=prod