For a matrix $A$ with orthogonal columns (but not necessarily normalized), the Gram matrix $G = A^T A$ satisfies $\det(G) = (\det(A))^2$. If $A$ is $3 \times 3$ with orthogonal columns of norms $\|\mathbf{c}_1\| = 2$, $\|\mathbf{c}_2\| = 3$, $\|\mathbf{c}_3\| = 4$, what is $\det(G)$?