3. If three non-zero and distinct real numbers $a, b, c$ satisfy $\frac{1}{a}+\frac{1}{b}=\frac{2}{c}$, then $a, b, c$ are called "harmonic"; if they satisfy $a+c=2b$, then $a, b, c$ are called "arithmetic".

Given the set $M=\{x|| x | \leqslant 2013, x \in \mathbf{Z}\}$, the set $P$ is a three-element subset of set $M$, i.e., $P=\{a, b, c\} \subset M$. If the elements $a, b, c$ in set $P$ are both harmonic and arithmetic, then the set $P$ is called a "good set".
The number of different good sets is $\qquad$
3.1006.

If $a, b, c$ are both harmonic and arithmetic, then
$$
\left\{\begin{array}{l}
\frac{1}{a} + \frac{1}{b} = \frac{2}{c}, \\
a + c = 2b
\end{array} \Rightarrow \left\{\begin{array}{l}
a = -2b \\
c = 4b
\end{array}\right.\right.
$$

Thus, the good set is of the form $\{-2b, b, 4b\}(b \neq 0)$.
Since the good set is a three-element subset of set $M$, we have
$$
\begin{array}{l}
-2013 \leqslant 4b \leqslant 2013(b \in \mathbf{Z}, \text{ and } b \neq 0) \\
\Rightarrow -503 \leqslant b \leqslant 503(b \in \mathbf{Z}, \text{ and } b \neq 0).
\end{array}
$$

Therefore, $b$ can take 1006 values.
Thus, the number of good sets is 1006.