A Column That Was Already There

A regression uses an intercept plus two engineered features. For four training samples the design matrix is

$X = \begin{pmatrix}1 & 1 & 3\\ 1 & 2 & 5\\ 1 & 3 & 7\\ 1 & 4 & 9\end{pmatrix}$

where column 1 is the intercept, column 2 is the raw feature, and column 3 was produced by a feature-engineering step.

The least-squares fit is found by solving the normal equations

$X^\top X\,\beta = X^\top y$

for some target vector $y \in \mathbb{R}^4$.

Give $\det(X^\top X)$, $\operatorname{rank}(X^\top X)$, and the number of solutions $\beta$ the normal equations admit.