2019 CIME I Problems/Problem 9

Let $\text{N}$ denote the number of strictly increasing sequences of positive integers $a_1,a_2,\cdots, a_{19}$ satisfying the following two rules $:$ $a_1=1$ and $a_{19}=361,$ for any $i \neq j,$ if $b_{ij}$ is the $(i \cdot j)^{\text{th}}$ number not in the sequence $,$ then $(a_i-b_{ij})(a_jb_{ij})<0.$ Find the largest positive integer $k$ such that $2^k$ divides $\text{N}.$