Prove that there's an uncountable family of subsets of $\mathbb N$ such that intersection of every 3 is infinite, and every 4 a finite set

Question

This is a question from my set theory exam

Prove that there's an uncountable family of subsets of $\mathbb N$ such that intersection of every 3 is infinite, and every 4 a finite set

EDIT: This question is a duplicate (Show the existence of family of sets)

There's similar questions, like this one, but my question seems significantly different

I have no idea even how to start, any hints would be appreciated