I'm trying to show that $\forall n \in \mathbb{N}$, some integer $z$ can be found satisfying: $$ 6n^2-2n \le z \lt 6(n+1)^2-2(n+1) $$ and $$ \prod_{k=1}^{n} \left( \left( z^2-k^2 \right) \bmod \left( 6k-1 \right) \right) \left( \left( z^2-k^2 \right) \bmod \left( 6k+1 \right) \right) \ne 0 $$
An example will probably help visualizing what I'm trying to achieve, let's take $n=2$
$$ \begin{array}{|c|c|c|c|c|} \hline z & \bmod 5 & \bmod 7 & \bmod 11 & \bmod 13 \\ \hline 20 & 0 & \color{red}{6} & \color{red}{9} & 7 \\ 21 & \color{red}{1} & 0 & 10 & 8 \\ 22 & 2 & \color{red}{1} & 0 & 9 \\ \color{green}{\fbox{23}} & 3 & 2 & 1 & 10 \\ 24 & \color{red}{4} & 3 & \color{red}{2} & \color{red}{11} \\ \end{array} $$ Marked in red are the zeros, and $23$ is the first non-zero product for the $n=2$ case.
So far, I've reached very little with this problem. I can say that non-prime moduli can be safely ignored. For instance, with $n=6$, all the zeros generated by the modulo 35 are already present in both moduli 5 and 7 columns. Also, any non-prime modulo is the product of moduli already present in the above formula (because the set of congruence classes $\left\{\overline{1}_6,\overline{5}_6\right\}$ is closed under multiplication). That is to say, if only considering the prime moduli helps obtaining a proof, then it's absolutely fine.
I'm quite unsure how to tackle the problem though. "Normal" modularity-related techniques don't seem to be very useful, but it wouldn't be the first time I'm overseeing something... Looking at it on a different angle, I noticed that for $n=1$, the longest stretch of rows with zeros seems to be of length $4$, which is far less than the $12n+4=16$ total length of the segment supposed to be covered, and for $n=2$, the longest I could come up with was $10$, again too short to possibly cover a span of length $28$. So maybe someone could solve this problem with some combinatorial technique, which is why I added that tag to the question.
Edit: I realize only now that this problem can be framed as an RNS one, so I'm adding another appropriate tag.
Edit (again): I came up with a strategy that might help solve the problem. The case $n=1$ can be reduced to a mapping of $\mathbb{Z}_5 \times \mathbb{Z}_7 \to \mathbb{Z}_{35}$:
$$ \begin{array}{|c|c|c|c|c|c|c|c|} \hline . & \overline{0}_7 & \overline{1}_7 & \overline{2}_7 & \overline{3}_7 & \overline{4}_7 & \overline{5}_7 & \overline{6}_7 \\ \hline \overline{0}_5 & 0 & \bbox[lightgrey,5px]{15} & 30 & 10 & 25 & 5 & \bbox[lightgrey,5px]{20} \\ \hline \overline{1}_5 & \bbox[lightgrey,5px]{21} & \bbox[lightgrey,5px]{1} & \bbox[lightgrey,5px]{16} & \bbox[lightgrey,5px]{31} & \bbox[lightgrey,5px]{11} & \bbox[lightgrey,5px]{26} & \bbox[lightgrey,5px]{6} \\ \hline \overline{2}_5 & 7 & \bbox[lightgrey,5px]{22} & 2 & 17 & 32 & 12 & \bbox[lightgrey,5px]{27} \\ \hline \overline{3}_5 & 28 & \bbox[lightgrey,5px]{8} & 23 & 3 & 18 & 33 & \bbox[lightgrey,5px]{13} \\ \hline \overline{4}_5 & \bbox[lightgrey,5px]{14} & \bbox[lightgrey,5px]{29} & \bbox[lightgrey,5px]{9} & \bbox[lightgrey,5px]{24} & \bbox[lightgrey,5px]{4} & \bbox[lightgrey,5px]{19} & \bbox[lightgrey,5px]{34} \\ \hline \end{array} $$
Essentially, $\mathbb{Z}_{35}$ is treated as a cyclic diagonal path (from top-left to bottom-right) along a torus. The question becomes: is there a stretch of $12n+4=16$ consecutive greyed cells along that cyclic diagonal path? Evidently not, in this particular case. In fact, in an attempt to simplify the problem, we can even ignore some of the non-greyed cells and just focus on the central rectangle marked by $2$ and $33$. Even in that new (and hopefully simpler) setting, the longest stretch of greyed cells would be only 8 units long.
For $n=2$, the same reasoning should apply with $\mathbb{Z}_5 \times \mathbb{Z}_7 \times \mathbb{Z}_{11} \times \mathbb{Z}_{13} \to \mathbb{Z}_{5005}$. The diagonal step is then $(+1,+1,+1,+1)$, the central block (a hyper-parallelepiped?) has dimensions $(2,4,6,8)$ and we should find that no stretches of greyed cells can be $12n+4=28$ units long. And so on.
I've tried researching the literature about residue number systems, but the vast majority of sources are only interested in computer science applications, and I've been unable so far to find the proper tools to make this proof. Please help!