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 \( S = \{ \lambda \in \mathbb{R} : \) the points \( A, B, C \) and \( D \) are coplanar }. Then \( \sum\limits_{\lambda \in S} (\lambda + 2)^{2} \) is equal to:
