Call a permutation $a_1, a_2, \ldots, a_n$ of the integers $1, 2, \ldots, n$ quasi-increasing if $a_k \leq a_{k+1} + 2$ for each $1 \leq k \leq n-1$. For example, $53421$ and $14253$ are quasi-increasing permutations of the integers $1$, $2$, $3$, $4$, $5$, but $45123$ is not. Find the number of quasi-increasing permutations of the integers $1$, $2$, $\ldots$, $7$.