Vector Algebra

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