fix: use background context for MCP session and add test

[Nerja]

The MCP session was being initialized using the request-scoped context passed to Tools(ctx). When this context was cancelled the underlying MCP connection was closed. However, the session remained cached in the mcptoolset struct, causing subsequent calls to fail.

Changed the Connect call in getSession to use context.Background().

Added TestSessionContextCancellation to verify that the context passed to connect is not cancelled.

For questions related to executing Ping as well to confirm the status of the connection, please see https://github.com/google/adk-go/issues/399#issuecomment-3643981984

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Summary of Changes

Hello @Nerja, I'm Gemini Code Assist[^1]! I'm currently reviewing this pull request and will post my feedback shortly. In the meantime, here's a summary to help you and other reviewers quickly get up to speed!

This pull request resolves a critical issue where the MCP session's connection was prematurely terminated due to its reliance on a request-scoped context. By switching to context.Background() for session initialization, the change ensures the session's longevity and prevents unexpected disconnections. A dedicated test case has been introduced to validate this corrected behavior.

Highlights

• MCP Session Context Fix: The MCP session initialization has been updated to use context.Background() instead of a request-scoped context. This resolves an issue where the underlying MCP connection would close prematurely if the request-scoped context was cancelled, while the session remained cached, leading to subsequent failures.
• New Test Case: A new test, TestSessionContextCancellation, has been added to verify that the context used for connecting to the MCP session is not inadvertently cancelled, ensuring the stability of the session.

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[dpasiukevich]

I don't think there's problem with request-scoped context. Because this context is only needed to create a session. Once session object is created even if the context is cancelled it doesn't affect the session. The MCP lib doesn't store the context.

@Nerja could you verify if this assumption is correct?

If it is, then setting context.Background() won't solve the issue. As we'd need rather to handle stale cached session object.

[Nerja]

Closing in favor of https://github.com/google/adk-go/pull/417 and based on https://github.com/google/adk-go/issues/399#issuecomment-3647187957