Suppose the function \psi satisfies \psi(1)=\sqrt{2+\sqrt{2+\sqrt{2}}} and \psi(3 x)+3 \psi(x)=\psi(x)^{3} for all real x. Determine the greatest integer less than \prod_{n=1}^{100} \psi\left(3^{n}\right).
Suppose the function \psi satisfies \psi(1)=\sqrt{2+\sqrt{2+\sqrt{2}}} and \psi(3 x)+3 \psi(x)=\psi(x)^{3} for all real x. Determine the greatest integer less than \prod_{n=1}^{100} \psi\left(3^{n}\right).