Primorial

For a positive integer $n$, the primorial $n\mathrm{\#}$ is defined as the product of all primes $\le n$: $2\mathrm{\#}=2,3\mathrm{\#}=6,6\mathrm{\#}=30,\dots$.

Define $F(n,p)=\{\begin{array}{ll}n\mathrm{\#}\mathrm{mod}p& ,\text{ if }n<p,\\ \frac{1}{p}\times (n\mathrm{\#})\mathrm{mod}p& ,\text{ if }n\ge p.\end{array}$

For example, $F(3,7)=2\times 3\mathrm{mod}7=6$ and $F(7,3)=2\times 5\times 7\mathrm{mod}3=1$.

For $P=998244353$, You are given $F(143,P)=675572163$ and $F({10}^9,P)=574463500$.

Find $F({10}^{13},P)$.

(My timing: ~20s for ${10}^{13}$, ~65s for ${10}^{14}$)