Let the position vectors of the points \(A, B, C\) and \(D\) be \(5 \hat{i}+5 \hat{j}+2 \lambda \hat{k}, \hat{i}+2 \hat{…

Let the position vectors of the points \(A, B, C\) and \(D\) 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 the set \(S=\{\lambda \in R\) : The points \(A, B, C\) and \(D\) are coplanar \(\}\). Then \(\sum_{\lambda \in S}(\lambda+2)^2\) is equal to

[JEE Main 2023, 6 Apr (Shift 1)]