
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 a function $f(x)$ with a domain of $\left(-\infty ,0\right)$⋃$\left(0,+\infty \right)$, which is monotonically increasing on $\left(-\infty ,0\right)$ and satisfies $f(ab)=f(a)+f(b)-1$ for any $a$ and $b$ in its domain. If there exists $x\in \left(1,+\infty \right)$ such that the inequality $f(mx)-f(\ln x) \gt f(-1)-1$ holds, then the range of real number $m$ is ______.

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