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

Let's think step by step and output the final answer within \boxed{}.
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