emphasis - Revolutions can, and often have, begun with reading - why "begun", why not "begin"?

Revolutions can, and often have, begun with reading.

I am curious to know about the use of the verb "begun" here. Though I am not saying it's wrong, but it can also be "begin". I see it this way. When deciding the form of verb in these kind of construction, I break the sentences and the choose the form of the verb.

For example -

I can, and definitely will, do it.

Consider the sentence. When I write the verb "do", I think of which form of "do" will be proper here. So I break the sentence - "I can do it" and "I will definitely do it". And in both the sentences the verb "do" is common. So I take the verb form "do" in "I can, and definitely will, do it".

But when it came to this sentence I can't use the above mentioned method to decide the verb.

Revolutions can, and often have, begun with reading. (This is quoted from an article)

Now if I try to break the sentence into two it comes down to - "Revolution can begin with reading" and "Revolution often have begun with reading". In both sentences two form is used - "begin" and "begun", so it gives the confusion which form of the verb we should use in the final sentence.

Please help.