
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 the function $f(x)=2\ln x$ $(\frac{1}{e}\leqslant x\leqslant e^{2})$, $g(x)=mx+1$, if there exist points on the graph of $f(x)$ and $g(x)$ that are symmetric about the line $y=1$, then the range of the real number $m$ is \_\_\_\_\_\_\_\_\_.

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