A set of three elements is called arithmetic if one of its elements is the arithmetic mean of the other two. Likewise, a set of three elements is called harmonic if one of its elements is the harmonic mean of the other two.

How many three-element subsets of the set of integers $\{z \in \mathbb{Z} \mid -2011 < z < 2011\}$ are arithmetic and harmonic?

**Remark:** The arithmetic mean $A(a,b)$ and the harmonic mean $H(a,b)$ are defined as:

\[A(a,b) = \frac{a+b}{2} \quad \text{and} \quad H(a,b) = \frac{2ab}{a+b} = \frac{2}{\frac{1}{a} + \frac{1}{b}},\]

respectively, where $H(a,b)$ is not defined for some $a$, $b$.

Let's think step by step and output the final answer within \boxed{}.
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