
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given that the function $f(x)$ is an odd function defined on $(-\infty, 0) \cup (0, +\infty)$, and $f(-1) = 0$. If for any two real numbers $x_1$, $x_2 \in (0, +\infty)$ and $x_1 \neq x_2$, the inequality $\frac{f(x_1) - f(x_2)}{x_1 - x_2} > 0$ always holds, determine the solution set of the inequality $xf(x) > 0$.

Present the answer in LaTex format: \boxed{Your answer}
user