Orthogonality & Least Squares · Superchallenge problem

In $\mathbb R^4$ let $W_1$ be the plane of vectors $(s, t, 0, 0)$ and $W_2 = \operatorname{span}\{(1, 1, 1, 1),\ (1, 2, 3, 4)\}$. Over all nonzero $\mathbf u \in W_1$ and nonzero $\mathbf v \in W_2$, find the largest possible value of $\cos^2\theta$, where $\theta$ is the angle between $\mathbf u$ and $\mathbf v$.