$\left|z_{1}\right|^{2}=1 \Rightarrow z_{1} \bar{z}_{1}=1,\left|z_{2}\right|^{2}=4 \Rightarrow z_{2} \bar{z}_{2}=4$

$\left|z_{3}\right|^{2}=9 \Rightarrow z_{3} \bar{z}_{3}=9$

$\left|9 z_{1} z_{2}+4 z_{1} z_{3}+z_{2} z_{3}\right|=12$

$\left|z_{3} \bar{z}_{3} z_{1} z_{2}+z_{2} \bar{z}_{2} z_{1} z_{3}+z_{1} \bar{z}_{1} z_{2} z_{3}\right|=12$

$\Rightarrow\left|z_{1}\right|\left|z_{2}\right|\left|z_{3}\right|\left|\bar{z}_{1}+\bar{z}_{2}+\bar{z}_{3}\right|=12$

$\Rightarrow\left|\bar{z}_{1}+\bar{z}_{2}+\bar{z}_{3}\right|=2 \Rightarrow\left|z_{1}+z_{2}+z_{3}\right|=2$