Let $K$ be the number of sequences $A_1,A_2,\dots ,A_n$ such that $n$ is a positive integer less than or equal to $9$, each $A_i$ is a subset of $1,2,3,4,5,6$, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. What is the remainder when $K$ is divided by $10$?