3. In $\triangle A B C$, the lengths of the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively. Point $G$ satisfies
$$
\overrightarrow{G A}+\overrightarrow{G B}+\overrightarrow{G C}=\mathbf{0}, \overrightarrow{G A} \cdot \overrightarrow{G B}=0 \text {. }
$$

If $(\tan A+\tan B) \tan C=m \tan A \cdot \tan B$, then $m=$ . $\qquad$
3. $\frac{1}{2}$.

It is known that $G$ is the centroid of $\triangle ABC$.
Since $G A \perp G B$, we have $a^{2}+b^{2}=5 c^{2}$.
Thus, $m=\frac{\sin ^{2} C}{\sin A \cdot \sin B \cdot \sin C}$
$$
=\frac{2 c^{2}}{a^{2}+b^{2}-c^{2}}=\frac{1}{2} \text {. }
$$