Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \ \vec{b} = 3\hat{i} + 2\hat{j} - \hat{k}, \ \vec{c} = \lambda \hat{j} + \mu \hat{k} \) and \( \hat{d} \) be a unit vector such that \( \vec{a} \times \hat{d} = \vec{b} \times \hat{d} \) and \( \vec{c} \cdot \hat{d} = 1 \). If \( \vec{c} \) is perpendicular to \( \vec{a} \), then \( |3\lambda \hat{d} + \mu \vec{c}|^{2} \) is equal to _________.
