$\Rightarrow \vec{a}+2 \overrightarrow{\mathrm{c}}=-2 \vec{b}$
$ \Rightarrow (\vec a + 2\vec c) \cdot (\vec a + 2\overrightarrow {\rm{c}} ) = ( - 2\vec b)( - 2\overrightarrow {\rm{b}} )$
$\Rightarrow \vec{a} \cdot \vec{a}+4 \vec{c} \cdot \vec{c}+4 \vec{a} \cdot \vec{c}=4 \vec{b} \cdot \vec{b}$
$\Rightarrow 1+4+4 \vec{a} \cdot \overrightarrow{\mathrm{c}}=4$
$\Rightarrow \vec{a} \cdot \vec{c}=\frac{-1}{4}$
$\because|\vec{a} \cdot \overrightarrow{\mathrm{c}}|^{2}+|\vec{a} \times \vec{c}|^{2}=1(\vec{a} \text { is unit vector })$
$\frac{1}{16}+|\bar{a} \times \bar{c}|^{2} =1 $
$|\vec{a} \times \vec{c}|^{2} =\frac{15}{16} $
$|\vec{a} \times \vec{c}| =\frac{\sqrt{15}}{4}$
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