
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 $f(x)$ on interval $A$, where $a$, $b$, $c \in A$, and $f(a)$, $f(b)$, $f(c)$ are the lengths of the sides of a triangle, then the function $f(x)$ is called a "triangle function". Given that the function $f(x) = x\ln x + m$ is a "triangle function" on the interval $\left[ \frac{1}{{e}^{2}} ,e\right]$, find the range of the real number $m$.

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