
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
Let functions $f(x)$ and $g(x)$ be defined on the same interval $[a,b]$. If for any $x \in [a,b]$, we have $|f(x)-g(x)| \leqslant 1$, then $f(x)$ and $g(x)$ are called "close functions" on $[a,b]$, and the interval $[a,b]$ is called a "close interval". If $f(x)=\ln x$ and $g(x)=\frac{mx-1}{x}$ are "close functions" on the interval $\left[\frac{1}{e},e\right]$, then the range of the real number $m$ is ______.

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