Upper bound of the modulus of a complex function

Question

Let $R>0$ be $\underline{\text{large}}$. If $0\leq t\leq \frac{\pi}{2}$, prove that $|\xi|\leq \frac{6}{R}$, where $\xi:=-\frac{2}{R}e^{-it}+\frac{4}{R^3}e^{-3it}$. As a first attempt, I can show that $|\xi|^2=\frac{4}{R^2}+\frac{16}{R^6}-\frac{16}{R^4}\cos(2t)$. However, I'm lost after this step. This is a part of a problem I found in a complex analysis book. Any help is appreciated.