Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k} , \\ \vec{b} = 2\hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{d} = \vec{a} \times \vec{b} \). If \( \vec{c} \) is a vector such that \( \vec{a}\cdot\vec{c} = |\vec{c}| \), \( |\vec{c} - 2\vec{a}|^{2} = 8 \) and the angle between \( \vec{d} \) and \( \vec{c} \) is \( \frac{\pi}{4} \), then \( |10 - 3\vec{b}\cdot\vec{c}| + |\vec{d}\times\vec{c}|^{2} \) is equal to ______.
