Let $\overrightarrow{\mathrm{a}}=2 \hat{\imath}-3 \hat{\jmath}+4 \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=3 \hat{\imath}+4 \hat{\jmath}-5 \hat{\mathrm{k}}$, and a vector $\vec{c}$ be such that $\vec{a} \times(\vec{b}+\vec{c})+\vec{b} \times \vec{c}=\hat{\imath}+8 \hat{\jmath}+13 \hat{k}$ If $\vec{a} \cdot \vec{c}=13$, then $(24-\vec{b} \cdot \vec{c})$ is equal to $\_\_\_\_$ .