(a) The complex numbers $z_{1}$ and $z_{2}$ satisfy the equations

$$z_{1}^{3}=1, \quad z_{2}^{9}=512 .$$

What are the possible values of $\left|z_{1}-z_{2}\right|$ ? Justify your answer.

(b) Show that $\left|z_{1}+z_{2}\right| \leqslant\left|z_{1}\right|+\left|z_{2}\right|$ for all complex numbers $z_{1}$ and $z_{2}$. Does the inequality $\left|z_{1}+z_{2}\right|+\left|z_{1}-z_{2}\right| \leqslant 2 \max \left(\left|z_{1}\right|,\left|z_{2}\right|\right)$ hold for all complex numbers $z_{1}$ and $z_{2}$ ? Justify your answer with a proof or a counterexample.