do I can't just sit and do nothing. (sit and do are the same time) or I can't just sit (here) doing nothing.

i cant just sit and do nothing and的左右两边都是要跟一样时态的词 can't的后面只能跟动词原型，所以是sit and do 希望对你有帮助

I'm agree with domasla

it 's not correct, because the "and" need two same form Verb on each side. I can't just sit and do nothing。