The sequence $(a_n)$ is defined recursively by $a_0=1,a_1=\sqrt[31]{7}$, and $a_n=a_{n-1}^6{a}_{n-2}^7$ for $n\ge 2$. What is the smallest positive integer $k$ such that the product $a_1{a}_2\cdots a_k$ is an integer?