For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which

$\left(a_1\right)^1 \times \left(a_2\right)^2 \times \cdots \times \left(a_{20}\right)^{20} < a_{21} \times a_{22} \times \cdots \times a_{(20+m)}$

is