Complex numbers $z_1, z_2$ simultaneously satisfy the conditions $|z_1+z_2|=|z_1z_2+1|$ and $3|z_1+1||z_2+1|=|z_1z_2+5(z_1+z_2)+1|$. Prove that $|z_1|+|z_2|=|z_1z_2|+1$.
I know $z_1=a+bi$ and $z_2=x+yi$ because they're complex, but after I made the calculation, nothing appeared to work. Any ideas?