Remember that began is used for actions that started and often completed in the past.
Begun is used with a helping verb like have or had to indicate something that started in the past and continues or just emphasizes the action's completion.
Think of began as standalone — it stands alone without any helper verbs.
In contrast, begun needs a helper horse like have to ride with it; it can't stand alone.
Began always walks solo, while begun needs have,has, or had for company.