Suppose the function $\psi$ satisfies $\psi(1) = \sqrt{2 + \sqrt{2 + \sqrt{2}}}$ and
\[\psi(3x) + 3 \psi(x) = \psi^3(x)\]for all real $x.$  Determine $\prod_{n = 1}^{100} \psi(3^n).$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Suppose the function $\psi$ satisfies $\psi(1) = \sqrt{2 + \sqrt{2 + \sqrt{2}}}$ and
\[\psi(3x) + 3 \psi(x) = \psi^3(x)\]for all real $x.$  Determine $\prod_{n = 1}^{100} \psi(3^n).$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
4096 \sqrt{2}
Suppose the function $\psi$ satisfies $\psi(1) = \sqrt{2 + \sqrt{2 + \sqrt{2}}}$ and
\[\psi(3x) + 3 \psi(x) = \psi^3(x)\]for all real $x.$  Determine $\prod_{n = 1}^{100} \psi(3^n).$
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