Construction of a bi-infinite power free word with a given factor and a non-recurrent letter

Let $L_{k,\alpha}^{\mathbb{Z}}$ denote the set of all bi-infinite $\alpha$-power free words over an alphabet with $k$ letters, where $\alpha$ is a positive rational number and $k$ is positive integer. We prove that if $\alpha\geq 5$, $k\geq 3$, $v\in L_{k,\alpha}^{\mathbb{Z}}$, and $w$ is a finite factor of $v$, then there are $\widetilde v\in L_{k,\alpha}^{\mathbb{Z}}$ and a letter $x$ such that $w$ is a factor of $\widetilde v$ and $x$ has only a finitely many occurrences in $\widetilde v$.