You're going to have to program that. Since there is no action you're responding to but rather an issue becoming stale, so you have no trigger to start the transition.
You could write a Dashboard widget that would show the issues you need to take action on but either a person or an external program would need to trigger the action.
I have multiple projects that use variations of the same base workflow. The variations depend on the requirements of the project or issue type. The variations mostly come in the form of new statuses ...
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