If the vectors \(\vec{a}=\lambda \hat{i}+\mu \hat{j}+4 \hat{k}, \vec{b}=-2 \hat{i}+4 \hat{j}-2 \hat{k}\) and \(\vec{c}=2…

If the vectors \(\vec{a}=\lambda \hat{i}+\mu \hat{j}+4 \hat{k}, \vec{b}=-2 \hat{i}+4 \hat{j}-2 \hat{k}\) and \(\vec{c}=2 \hat{i}+3 \hat{j}+\hat{k}\) are coplanar and the projection of \(\vec{a}\) on the vector \(\vec{b}\) is \(\sqrt{54}\) units, then the sum of all possible values of \(\lambda+\mu\) is equal to

[JEE Main 2023, 29 Jan (Shift 1)]