
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 $f(x)$ defined on $\mathbb{R}$ satisfying $f'(x) - f(x) = (1 - 2x)e^{-x}$ and $f(0) = 0$, determine which of the following statements are correct:
(write down the numbers of all correct statements)

(1) $f(x)$ has a maximum value but no minimum value;
(2) If the slope of the tangent lines at two different points A and B on the curve $f(x)$ is $k$, then the range of $k$ is $-\frac{1}{e^2} < k < 0$;
(3) For any $x_1, x_2 \in (2, +\infty)$, the inequality $f\left( \frac{x_1 + x_2}{2} \right) \leq \frac{f(x_1) + f(x_2)}{2}$ always holds;
(4) When $a \neq b$, the equation $f(a) = f(b)$ has exactly two pairs of different real solutions $(a, b)$ such that both $e^a$ and $e^b$ are integers.

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