low degree polynomial

let $\displaystyle \mathbb{F}$ be a finite field of size n, and let P be the set of all planes in $\displaystyle \mathbb{F}^3$ (meaning affine subspaces of dimension 2).
$\displaystyle S \subset P; \; \; |S|\geq\frac{2(d+1)}{n} |P|; \; \; 100d<n$. also A is a function that assigns to each s in S a degree d polynomial over s, such that if s, s' are two non parallel planes in S then $\displaystyle A(s)=A(s')$ on the intersection $\displaystyle s \cap s'$.

I need to show that there is a polynomial Q of degree 2d on $\displaystyle \mathbb{F}^3$ such that $\displaystyle Q\mid_{s}=A(s)$ for each $\displaystyle s\in S$

what I tried to do is take d parallel planes in S (which I can find using the pigeonhole principle) and use interpolation over them to find Q. now, for each plane s the isn't parallel to them, it intersect them in d*n points. if $\displaystyle deg(Q\mid_s)=d$ then I could use the Shwartz Zippel theorem to show that $\displaystyle Q\mid_s = A(s)$, but all I know is that the degree is at most 2d, and not d.

another idea, is to find 2d parallel plane, and to interpolate over d of them, and somehow show that the polynomial agrees on the other planes as well, but I don't know how to do that. the Q polynomial from above is d degree over all plane parallel to the planes that were in the interpolation, so maybe this could help