Let three vectors \( \vec{a}=\alpha \hat{i}+4\hat{j}+2\hat{k} \), \( \vec{b}=5\hat{i}+3\hat{j}+4\hat{k} \), \( \vec{c}=x\hat{i}+y\hat{j}+z\hat{k} \) form a triangle such that \( \vec{c}=\vec{a}-\vec{b} \) and the area of the triangle is \( 5\sqrt{6} \). If \( \alpha \) is a positive real number, then \( \left| \vec{c} \right|^{2} \) is
