Let $\vec a = \widehat {\rm{i}} + \widehat {\rm{j}} + \widehat {\rm{k}},\overrightarrow {\rm{b}} = 2\widehat {\rm{i}} + 2\widehat {\rm{j}} + \widehat {\rm{k}}$ and $\overrightarrow {\rm{d}} = \vec a \times \overrightarrow {\rm{b}} $ . If $\overrightarrow {\rm{c}} $ is a vector such that $\vec a \cdot \overrightarrow {\rm{c}} = |\overrightarrow {\rm{c}} |$, $|\overrightarrow {\rm{c}} - 2\vec a{|^2} = 8$ and the angle between $\overrightarrow {\rm{d}} $ and $\overrightarrow {\rm{c}} $ is $\frac{\pi }{4}$, then $|10 - 3\overrightarrow {\;{\rm{b}}} \cdot \overrightarrow {\rm{c}} | + |\overrightarrow {\rm{d}} \times \overrightarrow {\rm{c}} {|^2}$ is equal to_____.