3. Given $\triangle A B C$ where $A B=2, A C=1, B C=\sqrt{7}, O$ is the circumcenter of $\triangle A B C$, and $\overrightarrow{A O}=\lambda \overrightarrow{A B}+\mu \overrightarrow{A C}$, then the value of $\lambda+\mu$ is $\qquad$.
$$
\begin{array}{l}
\overrightarrow{B C}=\overrightarrow{A C}-\overrightarrow{A B} \Rightarrow 7=4+1-2 \overrightarrow{A B} \cdot \overrightarrow{A C} \Rightarrow \overrightarrow{A B} \cdot \overrightarrow{A C}=-1 . \\
\text { Then }\left\{\begin{array} { l } 
{ \vec { A O } \cdot \vec { A B } = \lambda \vec { A B } ^ { 2 } + \mu \vec { A B } \cdot \vec { A C } , } \\
{ \vec { A O } \cdot \vec { A C } = \lambda \vec { A B } \cdot \vec { A C } + \mu \vec { A C } }
\end{array} \Rightarrow \left\{\begin{array} { l } 
{ 2 = 4 \lambda - \mu , } \\
{ \frac { 1 } { 2 } = - \lambda + \mu }
\end{array} \Rightarrow \left\{\begin{array}{l}
\lambda=\frac{5}{6}, \\
\mu=\frac{4}{3}
\end{array} \Rightarrow \lambda+\mu=\frac{5}{6}+\frac{4}{3}=\frac{13}{6}\right.\right.\right. \text {. }
\end{array}
$$