Let \( \vec{a} = \hat{i} + 5\hat{j} + \alpha \hat{k} \), \( \vec{b} = \hat{i} + 3\hat{j} + \beta \hat{k} \) and \( \vec{c} = -\hat{i} + 2\hat{j} - 3\hat{k} \) be three vectors such that, \( \lvert \vec{b} \times \vec{c} \rvert = 5\sqrt{3} \) and \( \vec{a} \) is perpendicular to \( \vec{b} \). Then the greatest amongst the values of \( \lvert \vec{a} \rvert^2 \) is __________.
