
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
Given a function $f(x)$ with its derivative $f'(x)$, it satisfies the equation
$$ef(x) - f'(1)e^{x} + ef(0)x - \frac{1}{2}e x^{2} = 0.$$
(1) Find the explicit formula for $f(x)$.
(2) If the equation $f(x) - \frac{1}{2}x^{2} - m = 0$ has exactly two distinct real roots in the interval $[-1, 2]$, determine the range of the real number $m$.

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