For which real numbers $x$ do the vectors

$$(x, 1,1,1), \quad(1, x, 1,1), \quad(1,1, x, 1), \quad(1,1,1, x),$$

not form a basis of $\mathbb{R}^{4}$ ? For each such value of $x$, what is the dimension of the subspace of $\mathbb{R}^{4}$ that they span? For each such value of $x$, provide a basis for the spanned subspace, and extend this basis to a basis of $\mathbb{R}^{4}$.