AMC 12 2022a Problem 8

The infinite product $$\sqrt[3]{10} \cdot \sqrt[3]{\sqrt[3]{10}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{10}}} \cdots$$ evaluates to a real number. What is that number?

$\textbf{(A) }\sqrt{10} \textbf{(B) }\sqrt[3]{100} \textbf{(C) }\sqrt[4]{1000} \textbf{(D) }10 \textbf{(E) }10\sqrt[3]{10}$