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This is one of many ways. Debounce the switch and use it as the clock input to a counter, then decode the counter output for 5. Connect the counter's 6th output to it's reset.
I don't understand your question completely but it may help you to visualize the operation of the johnson counter circuit simulated on the Falstad demonstrator.
What started as a button press counter has somehow morphed into a timer of some sort involving minutes. Why don't you give us a complete description of what you're trying to accomplish?
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