`McpToolAdapter` Connects to the MCP Server in a Stateless Mode

**Bug Description**
I create an `AssistantAgent` that can use tools from the [Playwright MCP server](https://www.npmjs.com/package/@playwright/mcp). Then, I wrap the `AssistantAgent` into a `RoundRobinGroupChat` team to enable consecutive tool invocations. However, I find that the `AssistantAgent` can only successfully invoke the `browser_navigate` tool from the Playwright MCP server, but other tool invocations (e.g., `browser_save_as_pdf`) return **execution errors**. It seems that the bug stems from the following implementation: https://github.com/microsoft/autogen/blob/b62b12e3e7a67a4d60a978f0b76abe32d2869cba/python/packages/autogen-ext/src/autogen_ext/tools/mcp/_base.py#L65-L66 At each time the `AssistantAgent` invokes a MCP tool, the `McpToolAdapter` connects to the MCP server **from scratch** without maintaining previous connections.

**To Reproduce**
I use the following code to build an `AssistantAgent` and a `RoundRobinGroupChat` team.
```
import asyncio
from autogen_ext.tools.mcp import StdioServerParams, mcp_server_tools
from autogen_ext.models.openai import OpenAIChatCompletionClient
from autogen_agentchat.agents import AssistantAgent
from autogen_agentchat.teams import RoundRobinGroupChat
from autogen_agentchat.conditions import TextMentionTermination
from generation_security.autogen.web_surfer import load_model_client
import argparse
from autogen_agentchat.ui import Console


async def main() -> None:
    model_client=OpenAIChatCompletionClient(model="gpt-4")
    server_params = StdioServerParams(
        command="npx",
        args=["@playwright/mcp@latest", "--headless", "--executable-path", "/home/user/.cache/ms-playwright/chromium-1161/chrome-linux/chrome"],
    )
    server_params.read_timeout_seconds = 120
    tools = await mcp_server_tools(server_params)

    agent = AssistantAgent(
        name="WebSurfer",
        model_client=model_client,
        tools=tools,
        reflect_on_tool_use=False,
    )
    task_content = "Save the website https://nesa.zju.edu.cn/ as a PDF."
    team = RoundRobinGroupChat([agent], termination_condition=TextMentionTermination("TERMINATE"), max_turns=10)
    await Console(team.run_stream(task=task_content))


if __name__ == "__main__":
    asyncio.run(main())
```
Running the above code, I get the following results. The `browser_save_as_pdf` tool returns an error indicating the variable `this._page` created by the `browser_navigate` tool is missing. This means that each tool invocation by the `AssistantAgent` **reinitializes the MCP server**, making the variable `this._page` not consistently maintained in consecutive tool invocations.
```
---------- user ----------
Save the website https://nesa.zju.edu.cn/ as a PDF.
---------- WebSurfer ----------
[FunctionCall(id='call_8b5f31fc16db46daa2e863', arguments='{"url": "https://nesa.zju.edu.cn/"}', name='browser_navigate')]
---------- WebSurfer ----------
[FunctionExecutionResult(content='[TextContent(type=\'text\', text=\'\\n- Page URL: https://nesa.zju.edu.cn/\\n- Page Title: Index - NESA - Network System Security & Privacy Lab\\n- Page Snapshot\\n```yaml\\n- link [ref=s1e4]:\\n    - /url: index.html\\n- list [ref=s1e6]:\\n    - listitem [ref=s1e7]:\\n        - link "home" [ref=s1e8]:\\n            - /url: index.html\\n    - listitem [ref=s1e9]:\\n        - link "news" [ref=s1e10]:\\n            - /url: webpage/news.html\\n    - listitem [ref=s1e11]:\\n        - link "Research" [ref=s1e12]:\\n            - /url: webpage/research.html\\n    - listitem [ref=s1e13]:\\n        - link "publication" [ref=s1e14]:\\n            - /url: webpage/pubyear.html\\n    - listitem [ref=s1e15]:\\n        - link "Software" [ref=s1e16]:\\n            - /url: webpage/software.html\\n    - listitem [ref=s1e17]:\\n        - link "contact" [ref=s1e18]:\\n            - /url: webpage/contact.html\\n- heading "Welcome to NESA！" [level=1] [ref=s1e22]\\n- paragraph [ref=s1e23]: "The Network SystEm Security & PrivAcy (NESA) Research\\n    Lab, directed by Dr. Shouling Ji, works at the intersection of AI, network\\n    systems, data analytics, and information security fields. The lab seeks to\\n    develop theories, techniques, and systems to enable a more secure and\\n    efficient network infrastructure, with computer systems that are more\\n    accountable and less vulnerable to attacks and abuse. Our current research\\n    focuses on four areas: (1) Data-Driven Security, (2) AI and Security, (3)\\n    Software and System Security, and (4) Big Data Mining and Analysis."\\n- heading "News" [level=2] [ref=s1e32]\\n- paragraph [ref=s1e33]:\\n    - text: "03/30/2025 - Our paper \\\\"Beyond Static Pattern Matching? Rethinking\\n        Automatic Cryptographic API Misuse Detection in the Era of LLMs\\\\" was\\n        accepted by ISSTA 2025. Congrats to Yifan! 03/26/2025 - Our paper\\n        \\\\"Privacy-preserving Triangle Counting in Directed Graphs\\\\" was accepted\\n        by ICDE 2025. Congrats to Ziyao! 03/12/2025 - Our paper \\\\"TextDefense:\\n        Adversarial Text Detection based on Word Importance Score Dispersion\\\\"\\n        was accepted by IEEE Transactions on Dependable and Secure Computing\\n        (TDSC). Congrats to Lujia!"\\n    - link "More ......" [ref=s1e40]:\\n        - /url: ./webpage/news.html\\n- heading "Announcement" [level=2] [ref=s1e43]\\n- paragraph [ref=s1e44]: NESA Lab looks for highly motivated undergrads, grads,\\n    and postdocs to join the group. If you are interested in our research,\\n    please come to visit our lab and/or send us your resume.\\n- list [ref=s1e46]:\\n    - listitem [ref=s1e47]:\\n        - heading "Data-Driven Security" [level=3] [ref=s1e48]\\n        - paragraph [ref=s1e50]: Cyber Threat/Crime Mining and Analysis; Fraud Detection\\n            and Analysis; Medical Application/Data Security and Privacy;\\n            Password Security; and CAPTCHA Security.\\n        - link "read more" [ref=s1e51]:\\n            - /url: webpage/research.html\\n    - listitem [ref=s1e53]:\\n        - heading "AI Security" [level=3] [ref=s1e54]\\n        - paragraph [ref=s1e56]: Adversarial Learning; Machine Learning Security and\\n            Privacy; AI-aided Vulnerability Mining; Smart Fuzzing; and Secure\\n            Multi-Party Computing and Learning.\\n        - link "read more" [ref=s1e57]:\\n            - /url: webpage/research.html\\n    - listitem [ref=s1e59]:\\n        - heading "Privacy" [level=3] [ref=s1e60]\\n        - paragraph [ref=s1e62]: Graph Privacy; Anonymization and De-anonymization;\\n            Differential Privacy; and Web Privacy..\\n        - link "read more" [ref=s1e63]:\\n            - /url: webpage/research.html\\n    - listitem [ref=s1e65]:\\n        - heading "Big Data Analysis" [level=3] [ref=s1e66]\\n        - paragraph [ref=s1e68]: Social Network Computing and Analytics and Graph and\\n            Network Embedding.\\n        - link "read more" [ref=s1e69]:\\n            - /url: webpage/research.html\\n- paragraph [ref=s1e73]: Copyright - NESA All rights reserved.\\n```\\n\', annotations=None)]', name='browser_navigate', call_id='call_8b5f31fc16db46daa2e863', is_error=False)]
---------- WebSurfer ----------
[TextContent(type='text', text='\n- Page URL: https://nesa.zju.edu.cn/\n- Page Title: Index - NESA - Network System Security & Privacy Lab\n- Page Snapshot\n```yaml\n- link [ref=s1e4]:\n    - /url: index.html\n- list [ref=s1e6]:\n    - listitem [ref=s1e7]:\n        - link "home" [ref=s1e8]:\n            - /url: index.html\n    - listitem [ref=s1e9]:\n        - link "news" [ref=s1e10]:\n            - /url: webpage/news.html\n    - listitem [ref=s1e11]:\n        - link "Research" [ref=s1e12]:\n            - /url: webpage/research.html\n    - listitem [ref=s1e13]:\n        - link "publication" [ref=s1e14]:\n            - /url: webpage/pubyear.html\n    - listitem [ref=s1e15]:\n        - link "Software" [ref=s1e16]:\n            - /url: webpage/software.html\n    - listitem [ref=s1e17]:\n        - link "contact" [ref=s1e18]:\n            - /url: webpage/contact.html\n- heading "Welcome to NESA！" [level=1] [ref=s1e22]\n- paragraph [ref=s1e23]: "The Network SystEm Security & PrivAcy (NESA) Research\n    Lab, directed by Dr. Shouling Ji, works at the intersection of AI, network\n    systems, data analytics, and information security fields. The lab seeks to\n    develop theories, techniques, and systems to enable a more secure and\n    efficient network infrastructure, with computer systems that are more\n    accountable and less vulnerable to attacks and abuse. Our current research\n    focuses on four areas: (1) Data-Driven Security, (2) AI and Security, (3)\n    Software and System Security, and (4) Big Data Mining and Analysis."\n- heading "News" [level=2] [ref=s1e32]\n- paragraph [ref=s1e33]:\n    - text: "03/30/2025 - Our paper \\"Beyond Static Pattern Matching? Rethinking\n        Automatic Cryptographic API Misuse Detection in the Era of LLMs\\" was\n        accepted by ISSTA 2025. Congrats to Yifan! 03/26/2025 - Our paper\n        \\"Privacy-preserving Triangle Counting in Directed Graphs\\" was accepted\n        by ICDE 2025. Congrats to Ziyao! 03/12/2025 - Our paper \\"TextDefense:\n        Adversarial Text Detection based on Word Importance Score Dispersion\\"\n        was accepted by IEEE Transactions on Dependable and Secure Computing\n        (TDSC). Congrats to Lujia!"\n    - link "More ......" [ref=s1e40]:\n        - /url: ./webpage/news.html\n- heading "Announcement" [level=2] [ref=s1e43]\n- paragraph [ref=s1e44]: NESA Lab looks for highly motivated undergrads, grads,\n    and postdocs to join the group. If you are interested in our research,\n    please come to visit our lab and/or send us your resume.\n- list [ref=s1e46]:\n    - listitem [ref=s1e47]:\n        - heading "Data-Driven Security" [level=3] [ref=s1e48]\n        - paragraph [ref=s1e50]: Cyber Threat/Crime Mining and Analysis; Fraud Detection\n            and Analysis; Medical Application/Data Security and Privacy;\n            Password Security; and CAPTCHA Security.\n        - link "read more" [ref=s1e51]:\n            - /url: webpage/research.html\n    - listitem [ref=s1e53]:\n        - heading "AI Security" [level=3] [ref=s1e54]\n        - paragraph [ref=s1e56]: Adversarial Learning; Machine Learning Security and\n            Privacy; AI-aided Vulnerability Mining; Smart Fuzzing; and Secure\n            Multi-Party Computing and Learning.\n        - link "read more" [ref=s1e57]:\n            - /url: webpage/research.html\n    - listitem [ref=s1e59]:\n        - heading "Privacy" [level=3] [ref=s1e60]\n        - paragraph [ref=s1e62]: Graph Privacy; Anonymization and De-anonymization;\n            Differential Privacy; and Web Privacy..\n        - link "read more" [ref=s1e63]:\n            - /url: webpage/research.html\n    - listitem [ref=s1e65]:\n        - heading "Big Data Analysis" [level=3] [ref=s1e66]\n        - paragraph [ref=s1e68]: Social Network Computing and Analytics and Graph and\n            Network Embedding.\n        - link "read more" [ref=s1e69]:\n            - /url: webpage/research.html\n- paragraph [ref=s1e73]: Copyright - NESA All rights reserved.\n```\n', annotations=None)]
---------- WebSurfer ----------
[FunctionCall(id='call_d932494bde474f37ba6162', arguments='{}', name='browser_save_as_pdf')]
---------- WebSurfer ----------
[FunctionExecutionResult(content="Error: MCP tool execution failed: [TextContent(type='text', text='Error: Navigate to a location to create a page', annotations=None)]\n", name='browser_save_as_pdf', call_id='call_d932494bde474f37ba6162', is_error=True)]
---------- WebSurfer ----------
Error: MCP tool execution failed: [TextContent(type='text', text='Error: Navigate to a location to create a page', annotations=None)]
```

### Which packages was the bug in?

Python Extensions (autogen-ext)

### AutoGen library version.

0.4.9

### Model used

GPT-4

### Python version

Python 3.12

### Operating system

Ubuntu 20.04 LTS

Labels: help wanted, proj-extensions
State: closed