
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
If the function $f(x)$, for any $a$, $b$, $c$ in interval $A$, $f(a)$, $f(b)$, $f(c)$ can be the lengths of the sides of a triangle, then $f(x)$ is called a "triangle function". It is known that the function $f(x) = x\ln(x) + m$ is a "triangle function" on the interval $\left[\frac{1}{e^2}, e\right]$, then the range of the real number $m$ is \_\_\_\_\_\_.

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