Call a permutation $a_1, a_2, \cdots, a_n$ of the integers $1, 2, \cdots, n$ 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 $1, 2, \cdots, 7$.

Let's think step by step and output the final answer within \boxed{}.
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