Farey polygons I

Consider the Farey sequence of order n.

For instance $F_4=\{\frac{0}{0},\frac{0}{1},\frac{1}{4},\frac{1}{3},\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{1}{1}\}$
We define $F_n(z)$ by applying a zoom $z$ i.e. multiplying each term by $z$.

We consider each term of the sequence $F_n(z)$ as a point and join these points respecting the order.
We obtain the Farey polygon $P_n(z)$ of order $n$ and zoom $z$

$P_4(2)$ contains 5 points in its interior.

Find the first Farey polygon $P_n(2)$ whose inside contains (strict) more than 2000 lattice points.

Answer format: $n,count$


[My timing: 40 sec]