CN-TST 2023 Problem 6

Prove that: (1) In the complex plane, each line (except for the real axis) that crosses the origin has at most one point ${z}$ , satisfy $$\frac {1+z^{23}}{z^{64}}\in\mathbb R.$$ (2) For any non-zero complex number ${a}$ and any real number $\theta$ , the equation $1+z^{23}+az^{64}=0$ has roots in $$S_{\theta}=\left\{ z\in\mathbb C\mid\operatorname{Re}(ze^{-i\theta })\geqslant |z|\cos\frac{\pi}{20}\right\}.$$ Proposed by Yijun Yao