## 1. Backend - Proxy Endpoint

- [ ] 1.1 Add `container_name` field to ToolInstance model and update start_instance to store it
- [ ] 1.2 Create proxy endpoint `/instances/{id}/proxy/{path:path}` in tool_instances.py
- [ ] 1.3 Implement HTTP forwarding using httpx with streaming support
- [ ] 1.4 Add ownership check before proxying
- [ ] 1.5 Add WebSocket upgrade support for the proxy endpoint
- [ ] 1.6 Handle response header forwarding (Content-Type, cookies, etc.)

## 2. Backend - Instance URL Update

- [ ] 2.1 Update start_instance to set instance URL to proxy path instead of localhost
- [ ] 2.2 Ensure container_name is captured during start

## 3. Frontend - Update Instance Links

- [ ] 3.1 Update InstanceList "Open" button to use proxy URL
- [ ] 3.2 Update SessionsPage "Open" button to use proxy URL
- [ ] 3.3 Ensure URLs open in new tab

## 4. Testing & Quality

- [ ] 4.1 Test proxy with code-server instance
- [ ] 4.2 Verify WebSocket features work (terminal inside code-server)
- [ ] 4.3 Run quality gates (ruff, mypy, typecheck, lint, build)
- [ ] 4.4 Deploy and test end-to-end