$$
\operatorname{Re}\left(z_{1} \overline{z_{2}}\right)=\operatorname{Re}[(1+2 \mathrm{i})(m-3 \mathrm{i})]=m+6=0 \Rightarrow m=-6,
$$

So $\left|z_{1}+z_{2}\right|=|-5+5 \mathrm{i}|=5 \sqrt{2}$.
2. Let $m$ be a real number, and the complex numbers $z_{1}=1+2 \mathrm{i}, z_{2}=m+3 \mathrm{i}$ (where $\mathrm{i}$ is the imaginary unit). If $z_{1} \overline{z_{2}}$ is a pure imaginary number, then the value of $\left|z_{1}+z_{2}\right|$ is $\qquad$.