Investigate Github MCP tools in the context of the `run-gemini-cli` Github actions

[joshualitt]

There are two related subtasks to complete:

1. We should confirm that the Gemini CLI is able to work with the Github MCP tools effectively.
2. We should confirm the run-gemini-cli Github actions prompts work correctly non-interactively.

[joshualitt]

I've done a preliminary investigation, and so far I see no reason to suspect systemic issues with the Github MCP tools. However, this does not mean there are no such issues, but rather that there are other more basic issues we need to address first.

The major issues I've noticed so far are:

• Use of disallowed commands which probably should be allowed (pwd, and ls)
• Clear termination problems, i.e. model tries to exit 0, and then sleep to await the next instruction
• Agent demonstrating a lack of clarity on lifecycle.
  • Responding textually to the user instead of using the Github MCP tools
  • "Waiting" for user input by quiescing(and thus exiting Gemini CLI)
  • Inappropriately using shell tools when the github tools are provided
  • The general termination issues may be a symptom of this
• Gemini can flail when difficulties are encountered during a task, which can quickly introduce a considerable amount of context rot
• A number of actions don't appear to be filtering tools from the Github MCP server, and as a result they will serve up 101 tools and 2 prompts, most of which aren't relevant.
• At least one tool name may have been outdated(created PR Fix incorrect tool name in gemini-invoke and add a few missing tools. google-github-actions/run-gemini-cli#359 to address this, and I audited the other tool names used, but this is something that should be verified by a test)

Given these issues, and their severity, I'm going to close this issue now in favor of focusing on observability. We can revisit the question of the performance of the Github MCP tools until after we've tackled these other issues.

Note: some of the agent lifecycle problems could be a result of conflicting instructions between the Gemini CLI's system prompt and the Github actions pseudo-system prompts. We should carefully audit the Gemini CLI's system prompt, and potentially consider allowing users to substitute some or all of the system prompt for cases where the agent is being embedded in a dramatically different environment.