\[\Large a_0 = 1, a_1 = \sqrt[19]{2}, \\ \Large a_n = a_{n-1} \times a_{n-2} ^2 \]

Consider a sequence defined recursively for \( n\geq 2 \) in the above manner.

What is the smallest positive integer value \(k\) such that the product \(a_{1}a_{2}\dotsm a_k\) is an integer?

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