Let \( \vec{a}=\hat{i}-\alpha\hat{j}+\beta\hat{k},\ \vec{b}=3\hat{i}+\beta\hat{j}-\alpha\hat{k} \) and \( \vec{c}=-\alpha\hat{i}-2\hat{j}+\hat{k} \), where \( \alpha \) and \( \beta \) are integers. If \( \vec{a}\cdot\vec{b}=-1 \) and \( \vec{b}\cdot\vec{c}=10 \), then \( (\vec{a}\times\vec{b})\cdot\vec{c} \) is equal to __________.
