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 \( \vec{a} \times (\vec{b} \times \vec{c}) + \vec{b} \times \vec{c} = \hat{i} + 8\hat{j} + 13\hat{k} \). If \( \vec{a} \cdot \vec{c} = 13 \), then \( \left(24 - \vec{b} \cdot \vec{c}\right) \) is equal to ________.
