[FEATURE]: Real-Time Streaming for AI Bot Debate Responses via SSE

### Feature and its Use Cases

#### Improvement
- **Current Behavior**: When the user finishes their turn, the application blocks and waits 3–6 seconds for the entire Gemini LLM payload to complete before rendering the full paragraph all at once.
- **Goal**: Render the bot's response word-by-word with zero initial latency as tokens arrive from the AI model, providing clear visual feedback (typing indicators and blinking cursor).

### Additional Context

Hi @raghavpuri31 i want to work on this feature .

### Code of Conduct

- [x] I have joined the [Discord server](https://discord.gg/hjUhu33uAn) and will post updates there
- [x] I have searched existing issues to avoid duplicates

The issue names no files, tests, or entry points, so start by tracing the current turn-completion path and how the Gemini payload reaches the UI. Done means responses arrive incrementally through SSE with no initial wait, while the interface shows typing feedback and a blinking cursor.

索引モデルが issue の本文から書いたものです。