Let \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k}, \ \vec{b} = 3\hat{i} - 3\hat{j} + 3\hat{k}, \ \vec{c} = 2\hat{i} - \hat{j} + 2\hat{k} \) and \( \vec{d} \) be a vector such that \( \vec{b} \times \vec{d} = \vec{c} \times \vec{d} \) and \( \vec{a} \cdot \vec{d} = 4 \). Then \( \left| \left( \vec{a} \times \vec{d} \right) \right|^{2} \) is equal to __________.
