A $3$-smooth number is an integer which has no prime factor larger than $3$. For an integer $N$, we define $S(N)$ as the set of $3$-smooth numbers less than or equal to $N$. For example, $S(20)=\{1,2,3,4,6,8,9,12,16,18\}$.

We define $F(N)$ as the number of permutations of $S(N)$ in which each element comes after all of its proper divisors.

This is one of the possible permutations for $N=20$.
- $1,2,4,3,9,8,16,6,18,12.$
This is not a valid permutation because $12$ comes before its divisor $6$.
- $1,2,4,3,9,8,\mathbf{12},16,\mathbf{6},18$.

We can verify that $F(6)=5$, $F(8)=9$, $F(20)=450$ and $F(1000)\approx 8.8521816557\mathrm{e}21$.
Find $F({10}^{18})$. Give as your answer its scientific notation rounded to ten digits after the decimal point.
When giving your answer, use a lowercase e to separate mantissa and exponent. E.g. if the answer is $112233445566778899$ then the answer format would be 1.1223344557e17.