Given small parameters $0<\epsilon<\epsilon'$ is there an $n_\epsilon>0$ such that at every $n>n_\epsilon$ if we are given a prime $n^2<p<2n^2$ then can we always find integers $a,b,c,d,e,f$ with $a,b$ coprime and $n^{1+\epsilon}<a,b<2n^{1+\epsilon}$ with $a$ even and $b,e,f$ odd, $p/2<c,d<p$ and $p^{\epsilon'}<e,f<2p^{\epsilon'}$ such that $$p|(ad-bc)\mbox{ and }p|(cf-de)$$ holds?
$$\mbox{Note }ad\equiv bc\bmod p,\quad cf\equiv de\bmod p\implies cd^{-1}\equiv ab^{-1}\equiv ef^{-1}\bmod p\mbox{ holds}.$$
If $e,f$ odd are fixed with $p^{\epsilon'}<e,f<2p^{\epsilon'}$ then there are $\frac{\frac{n^{1+\epsilon}}2n^{1+\epsilon}}{\zeta(2)}=\frac{n^{2(1+\epsilon)}}{2\zeta(2)}$ choices of coprime $a,b$ with $a$ even and so at least one of these choices should give right $a,b$ with $ab^{-1}\equiv ef^{-1}\bmod p$ (we should expect $\frac{n^{2\epsilon}}{2\zeta(2)}$ choices of coprime $a,b$ with $a$ even since $p$ is of size $O(n^2)$). A similar argument holds for $c,d$.
Do such sextuples really exist? The argument indicates each of the $O(p^{2\epsilon'})$ different pairs of $e,f$ have at least one $a,b,c,d$ associated with them.