Simplify arbitrary union expression in set theory

Let $A_k$ be the the set $A_k=\left\{x\in\left.\mathbb{R}\right|x^2-\left(4k+1\right)x+4k^2+2k<0\right\}$

I need to find a simplified way to rewrite the following group : $\mathbb{R}\setminus\left(\bigcup_{k\in z} A_k\right)$

By simplified I mean - not using arbitrary union or arbitrary intersection

The only equality I could think of is the following: $$\mathbb{R}\setminus\left(\bigcup_{k\in z} A_k\right) = \bigcup_{k\in z}\left[2k+1,2k+2\right]$$
But I still had to use arbitrary union, which I can't.

I would appriciate your assitance with finding the answer and prove the equality.