Let \( \vec{a} = -\hat{i} + 2\hat{j} + 2\hat{k} \), \( \vec{b} = 8\hat{i} + 7\hat{j} - 3\hat{k} \) and \( \vec{c} \) be a vector such that \( \vec{a} \times \vec{c} = \vec{b} \). If \( \vec{c} \cdot (\hat{i} + \hat{j} + \hat{k}) = 4 \), then \( \left|\vec{a} + \vec{c}\right|^2 \) is equal to:
