Problem: Let $z_1,$ $z_2,$ $z_3$ be complex numbers such that $|z_1| = 1,$ $|z_2| = 2,$ $|z_3| = 3,$ and $$|9z_1 z_2 + 4z_1 z_3 + z_2 z_3| = 12.$$Find $|z_1 + z_2 + z_3|.$
Progress: I received this problem yesterday as a challenge, but I cannot seem to find the answer. I know that brute-forcing my way through will probably be impossible, as it gets really messy really quickly. I am noticing that if I square the magnitude equations, I get $z_1\overline{z_1}=1$, $z_2\overline{z_2}=4$, and $z_3\overline{z_3}=9$, the coefficients of the long equation, hence why the title is as it is. From here, however, I am not sure how to continue, can somebody please give a solution explaining your steps? Thanks! (I'm new to this, so if I'm doing anything wrong, please let me know as well.)
