
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
Define: A function $f(x)$ is called a "mean value function" on the interval $[a, b]$ if there exist $x_1, x_2$ ($a < x_1 < x_2 < b$) such that:
$$
f'(x_{1}) = f'(x_{2}) = \frac{f(b) - f(a)}{b - a}
$$
Given the function $f(x) = \frac{1}{3}x^3 - \frac{1}{2}x^2 + m$ is a "mean value function" on the interval $[0, m]$, find the range of the real number $m$.

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