Let \( \vec{x} \) be a vector in the plane containing vectors \( \vec{a}=2\hat{i}-\hat{j}+\hat{k} \) and \( \vec{b}=\hat{i}+2\hat{j}-\hat{k} \). If the vector \( \vec{x} \) is perpendicular to \(( 3\hat{i}+2\hat{j}-\hat{k} )\) and its projection on \( \vec{a} \) is \( \dfrac{17\sqrt{6}}{2} \), then the value of \( |\vec{x}|^{2} \) is equal to ________.
