Two lines are described by the parametric equations:
\[
\mathbf{r}_1(t) = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + t \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}
\]
\[
\mathbf{r}_2(s) = \begin{pmatrix} 4 \\ 0 \\ -2 \end{pmatrix} + s \begin{pmatrix} 1 \\ 1 \\ k \end{pmatrix}
\]
Find the value of \( k \) if these lines intersect. Let's think step by step and output the final answer within \boxed{}.
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