$(3 \vec{a}+\vec{b}) \cdot(3 \vec{a}+\vec{b})=(2 \vec{a}+3 \vec{b}) \cdot(2 \vec{a}+3 \vec{b})$
$9 \vec{a} \cdot \vec{a}+6 \vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{b}=4 \vec{a} \cdot \vec{a}+12 \vec{a} \cdot \vec{b}+9 \cdot \vec{b} \cdot \vec{b}$
$5|\vec{a}|^{2}-6 \vec{a} \cdot \vec{b}=8|\vec{b}|^{2}$
$5(8)^{2}-6.8 .|\vec{b}| \cos 60^{\circ}=8|\vec{b}|^{2}$ $[\frac{1}{8}|\vec{a}|=1\Rightarrow|\vec{a}|=8]$
$40-3|\overrightarrow{\mathrm{b}}|=|\overrightarrow{\mathrm{b}}|^{2}$
$\Rightarrow|\overrightarrow{\mathrm{b}}|^{2}+3|\overrightarrow{\mathrm{b}}|-40=0$
$|\overrightarrow{\mathrm{b}}|=-8, \quad|\overrightarrow{\mathrm{b}}|=5$
$\quad$(rejected)
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