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Let $\vec{e}_{1}, \vec{e}_{2}$, and $\vec{e}_{3}$ be the standard basis for $\mathbb{R}^{3}$ and let $L: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ be a linear transformation with the properties
$L\left(\vec{e}_{1}\right)=\vec{e}_{2}, \quad L\left(\vec{e}_{2}\right)=2 \vec{e}_{1}+\vec{e}_{2}, \quad L\left(\vec{e}_{1}+\vec{e}_{2}+\vec{e}_{3}\right)=\vec{e}_{3} .$
Find a vector $\vec{v}$ such that $L(\vec{v})=k \vec{v}$ for some real number $k$.
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