The sequence \((a_n)\) is defined recursively by \(a_0=1\), \(a_1=\sqrt[19]{2}\), and \(a_n=a_{n-1}a_{n-2}^2\) for \(n\geq 2\). Find the smallest positive integer \(k\) such that the product \(a_1a_2\cdots a_k\) is an integer.

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