Let $\vec{u} = u_1 \hat{i} + u_2 \hat{j}$ be a unit vector in the $xy$ plane and $\vec{w} = \frac{1}{\sqrt{6}} (\hat{i} + \hat{j} + 2\hat{k})$. Given that there exists a vector $\vec{v}$ in $\mathbb{R}^3$ such that $|\vec{u} \times \vec{v}| = 1$ and $\vec{w} \cdot (\vec{u} \times \vec{v}) = 1$, then