Let \(\vec{u} = u_1 \hat{i} + u_3 \hat{k}\) be a unit vector in the xz-plane and \(\vec{q} = \frac{1}{\sqrt{6}} (\hat{i} + \hat{j} + 2\hat{k})\). If there exists a vector \(\vec{c}\) such that \(|\vec{u} \times \vec{c}| = 1\) and \(\vec{q} \cdot (\vec{u} \times \vec{c}) = 1\), then