
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
Suppose that for any function $f(x)$ defined on an interval $A$, and for all $a, b, c \in A$, $f(a), f(b), f(c)$ represent the lengths of the sides of a triangle. In this case, we call $f(x)$ 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 values for the real number $m$.

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