Does a Context-free language have a grammar that has either 3 or 0 nonterminals on the right hand side? Is the following true or false? Why?

Let L be a context-free language with $\epsilon\notin$ L. Then there is $\epsilon$-free grammar $G=(V,\Sigma, P,S )$ with $L (G) = L$, so all production rules are of form $A \to BCD$ or $A\to a$ with $A, B, C, D \in V$ and $a\in$ $\Sigma$.

I don't know normal forms for context-free languages. Just the CNF for regular ones. Trying a few small examples don't gives me a generally valid response, if there are really always production rules of this form.
Hint, can the production rules of that form generate a word  of length 2? Is there a context-free language that contains a word of length 2?