Support cookies in built-in browser

### Feature summary

_No response_

### What problem are you trying to solve?

When iterating on an app locally that needs authentication (for example, via OAuth), it is really annoying that I have to go through the authentication every time I relaunch the browser tab. It would be better if that tab supports cookies so that I can continue to share it with Copilot without the added friction of logging in every time.

### Proposed solution

_No response_

### Workflow impact

_No response_

### Installation context

_No response_

### Additional context

_No response_

首先定位内置浏览器和标签页的实现，然后检查当前如何处理跨重新启动的身份验证或会话状态。定义使用 OAuth 的本地应用所需的 Cookie 生命周期和持久化行为。完成标准是重新启动后仍保留相关登录状态，无需再次进行身份验证，并且为报告的工作流提供覆盖。

由索引模型根据 Issue 内容生成。