
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
When $f(x)$ is a differentiable function on $\mathbb{R}$ and its graph is continuous, with $f'(x) + x^{-1}f(x) > 0$ for $x \neq 0$, determine the number of zeros of the function $g(x) = f(x) + x^{-1}$.

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