If the vectors \(\vec{a}=\lambda \overset{^}{i}+\mu \overset{^}{j}+4\overset{^}{k},\vec{b}=−2\overset{^}{i}+4\overset{^}…

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

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