12.3.1.5 Algebraic

Problem: Find the maximum value of the parameter \( a \), for which the system has a solution: \[ \left\{\begin{array}{l} 4 \sin ^{2} y-a=16 \sin ^{2} \frac{2 x}{7}+9\left(\tan ^{-1}\right) \frac{2 x}{7} \\ \left(\pi^{2} \cos ^{2} 3 x-2 \pi^{2}-72\right) y^{2}=2 \pi^{2}\left(1+y^{2}\right) \sin 3 x \end{array}\right. \]