Mathematics Question from JEE Main 2023 · 29 January, Shift 1

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