Orthogonality & Least Squares · Superchallenge problem

For a nonzero vector $\mathbf w \in \mathbb R^6$ let $H_{\mathbf w} = I - \dfrac{2\,\mathbf w\mathbf w^{T}}{\mathbf w^{T}\mathbf w}$ be the reflection across the hyperplane orthogonal to $\mathbf w$. With $\mathbf u = (1, 0, 0, 0, 0, 0)$ and $\mathbf v = (1, 1, 1, 1, 0, 0)$, find $\operatorname{tr}\big((H_{\mathbf u}H_{\mathbf v})^{2026}\big)$.