Question:

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