JEE 2023 — Mathematics PYQ

Let the position vectors of the points $A,B,C$ and $D$ be $5\mathbf{i}+5\mathbf{j}+2\lambda \mathbf{k},\mathbf{i}+2\mathbf{j}+3\mathbf{k},-2\mathbf{i}+\lambda \mathbf{j}+4\mathbf{k}$ and  $-\mathbf{i}+5\mathbf{j}+6\mathbf{k}$. Let the set $S=\{\lambda \in \mathbb{R}:\text{the points }A,B,C\text{ and }D\text{ are coplanar}\}$. Then $\sum _{\lambda \in S}(\lambda +2{)}^2$ is equal to