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Ok. I have discovered the source of our confusion. We are thinking of two different verbs here.

I am talking about running in general. The perfective forms are побегать/побежать.

они побежали...

Prefixes! Of course! Thanks for catching that.

I think we are having a little dictionary conflict here. The "501 Russian Verbs" book says that побегать/побежать is perfective. Maybe someone...

You can add like tons of endings to motion verbs. It's just that in the dictionary, perfective forms are listed with the prefix по-. по also means "to set off".

они побежали домой. - They ran...

Aren't the perfective forms пoбегать/побежать?