Have nothing to do ~

Jason's boss helped him with the work so much that he nearly had nothing to do but handed in the completed work .

could I say "to hand in" instead of " handed in" in the above sentence?

what kind of verb forms should follow the expression " have nothing to but~"? A verb root, to infinitive and a past tense verb could come after the expression " have nothing to do but~", as follows?

I have nothing to do but call him.
I have nothing to do but to call him.
I had nothing to do but called him.