Absorbing Man and Exhaust

I have an \[\[Absorbing Man\]\] EDH deck and had a question about how he interacts with Exhaust effects.

If I have an \[\[Absorbing Man\]\] and a \[\[Riverchurn Monument\]\] on the field and have Absorbing Man become a copy of the monument until the beginning of my next turn. Then I use the Exhaust ability of the monument. Then the following turn I choose to make Absorbing Man a copy of the Monument again.

Can I activate the Exhaust ability again, or is it still considered an exhausted ability of the same permanent even though it lost and then regained the ability?

Thanks!