Background Bash system-reminders persist after process completion and output retrieval

Description

When using run_in_background: true with the Bash tool, system-reminders about background processes persist indefinitely even after:

1. The process has completed (exit_code=0)
2. Output has been retrieved via BashOutput
3. KillShell has been attempted (correctly reports "Shell is not running, status: completed")

The reminders appear with every subsequent tool response for the remainder of the session.

Steps to Reproduce

1. Run two Bash commands with run_in_background: true:

Bash("python script.py", run_in_background=true)  # ID: 46410c
Bash("python script2.py", run_in_background=true) # ID: 456b2e

1. Wait for processes to complete

1. Retrieve output with BashOutput:

BashOutput(bash_id="46410c")  # Returns status: completed, exit_code: 0
BashOutput(bash_id="456b2e")  # Returns status: completed, exit_code: 0

1. Attempt cleanup with KillShell:

KillShell(shell_id="46410c")  # Returns: "Shell is not running, status: completed"
KillShell(shell_id="456b2e")  # Returns: "Shell is not running, status: completed"

1. Make any subsequent tool call

Expected Behavior

System-reminders should stop appearing after:

• Process completes AND output is retrieved via BashOutput, OR
• KillShell is called on a completed process

Actual Behavior

System-reminders continue appearing with every tool response:

<system-reminder>
Background Bash 46410c (command: ...) (status: running) Has new output available. 
You can check its output using the BashOutput tool.
</system-reminder>

<system-reminder>
Background Bash 456b2e (command: ...) (status: running) Has new output available.
You can check its output using the BashOutput tool.
</system-reminder>

Issues with the reminders:

1. Status shows "running" when process is actually "completed"
2. Reminders persist after output has been read
3. Reminders persist after KillShell attempt
4. No way to dismiss/clear them

Impact

• Token waste: Each reminder consumes ~50 tokens, repeated with every tool call
• Context pollution: Stale reminders clutter the conversation
• User confusion: Status shows "running" for completed processes
• Workaround required: Users must avoid run_in_background entirely

Environment

• Claude Code CLI
• Windows 11
• Model: claude-opus-4-5-20250514

Suggested Fix

The background bash tracker should:

1. Mark processes as "acknowledged" after BashOutput retrieves completed output
2. Stop generating reminders for acknowledged/completed processes
3. Show accurate status ("completed" not "running") in reminder text
4. Provide a way to explicitly dismiss reminders (e.g., AcknowledgeBash tool)