fix(mcp): mcp client session recreates every time per tool calling

[BoyuanZhangDE]

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Link to Issue or Description of Change

1. Link to an existing issue (if applicable):

• Closes: ADK LlmAgent Does Not Persist MCPToolset Session Across Tool Calls, Causing Browser Context to Close After Each Action #2927
• Related: fix: Enhance error handling for closed and broken resources in retry … #2929

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Response from ADK Triaging Agent

Hello @BoyuanZhangDE, thank you for creating this PR!

To help reviewers understand and review your contribution more efficiently, could you please fill out the PR description template? Specifically, please provide:

• A link to the issue this PR is addressing or a description of the change.
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[hangfei]

@boyuan-zhang-ck Could you sign the CLA so we can start the review? Thanks.

[BoyuanZhangDE]

@boyuan-zhang-ck Could you sign the CLA so we can start the review? Thanks.

Hi! @hangfei So sorry for the delay. I just signed the CLA.
If possible, would you mind delaying review for another week? Unfortunately, I had some work changes recently and just started to have time for this PR now. This PR still needs some local testings to be ready for review.

[jnuvola]

@BoyuanZhangDE @hangfei
I noticed this PR and wanted to check in as I'm also affected by this. Is there any progress or plan for moving this forward? Happy to help or provide more info if needed.