
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 $f(x)$ has a unique $x (a < x < b)$ on the interval $[a,b]$ that satisfies $(b-a) \cdot f'(x) = f(b) - f(a)$, then the function $f(x)$ is called a "single-valued function" on $[a,b]$. Given that the function $f(x) = x^3 - x^2 + m$ is a "single-valued function" on $[0,a]$ $(a > \frac{1}{2})$, when the real number $a$ takes the minimum value, and the function $f(x)$ has exactly two zeros on $[0,a]$, the range of the real number $m$ is \_\_\_\_\_\_.

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