ci: add GitHub Action to build and push Gemini proxy image

Automatically builds the runpod-proxy Docker image and pushes to
GitHub Container Registry when changes are made to runpod-proxy/

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

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

.github/workflows/build-proxy.yml

@@ -0,0 +1,46 @@
+name: Build Gemini Proxy
+
+on:
+  push:
+    paths:
+      - 'runpod-proxy/**'
+  workflow_dispatch:
+
+env:
+  REGISTRY: ghcr.io
+  IMAGE_NAME: ${{ github.repository }}/gemini-proxy
+
+jobs:
+  build-and-push:
+    runs-on: ubuntu-latest
+    permissions:
+      contents: read
+      packages: write
+
+    steps:
+      - name: Checkout repository
+        uses: actions/checkout@v4
+
+      - name: Log in to Container Registry
+        uses: docker/login-action@v3
+        with:
+          registry: ${{ env.REGISTRY }}
+          username: ${{ github.actor }}
+          password: ${{ secrets.GITHUB_TOKEN }}
+
+      - name: Extract metadata
+        id: meta
+        uses: docker/metadata-action@v5
+        with:
+          images: ${{ env.REGISTRY }}/${{ env.IMAGE_NAME }}
+          tags: |
+            type=raw,value=latest
+            type=sha
+
+      - name: Build and push
+        uses: docker/build-push-action@v5
+        with:
+          context: ./runpod-proxy
+          push: true
+          tags: ${{ steps.meta.outputs.tags }}
+          labels: ${{ steps.meta.outputs.labels }}