Tessellate S.T.E.M.S - Mathematics - School - Set 1 - Problem 5

We call an integer $k \geq 5$ good if there exists integers $a,b,$ such that $a^2+k= b^2+2k=c^2+3k$ . Which of the following options are correct?

$(a)$ There exists finitely many good integers of the form $4k,\hspace{3pt} k \in \mathbb{N}$ and infinitely many good integers of the form $4k+1 ,\hspace{3pt} k \in \mathbb{N}$

$(b)$ There exists infinitely many good integers of the form $4k,\hspace{3pt} k \in \mathbb{N}$ and no good integers of the form $4k+1 ,\hspace{3pt} k \in \mathbb{N}$

$(c)$ There exists no good integers of the form $4k,\hspace{3pt} k \in \mathbb{N}$ and finitely many good integers of the form $4k+1 ,\hspace{3pt} k \in \mathbb{N}$

$(d)$ None of the above