Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \), \( \vec{b} = -\hat{i} - 8\hat{j} + 2\hat{k} \) and \( \vec{c} = 4\hat{i} + c_2 \hat{j} + c_3 \hat{k} \) be three vectors such that \( \vec{b} \times \vec{a} = \vec{c} \times \vec{a} \). If the angle between vector \( \vec{c} \) and the vector \( 3\hat{i} + 4\hat{j} + \hat{k} \) is \( \theta \), then the greatest integer less than or equal to \( \tan^2 \theta \) is \( \_\_\_\_\_\_\_\_\_\_ \).
