\(\begin{aligned} &(\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2=676 \\ &(|\vec{a}| \cdot|\vec{b}| \sin \theta \hat{n})^2+(|\vec{a}| \cdot|\vec{b}| \cos \theta)^2=676 \\ & \Rightarrow a^2 b^2 \sin ^2 \theta+a^2 b^2 \cos ^2 \theta=676\left[(\hat{n})^2=1\right] \\ & a^2 b^2\left(\sin ^2 \theta+\cos ^2 \theta\right)=676 \Rightarrow a^2=\frac{676}{b^2}=\frac{676}{4} \\ &|\vec{a}|=\sqrt{\frac{676}{4}} \Rightarrow|\vec{a}|=\frac{26}{2} \Rightarrow|\vec{a}|=13\end{aligned}\)