
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
Let $f(x) = x^3 + bx^2 + cx + d$, and let $k$ be a constant. It is known that when $k < 0$ or $k > 4$, the equation $f(x) - k = 0$ has only one real root; when $0 < k < 4$, the equation $f(x) - k = 0$ has three distinct real roots. Consider the following statements:  
(1) $f(x) - 4 = 0$ and $f'(x) = 0$ have one real root in common;  
(2) $f(x) = 0$ and $f'(x) = 0$ have one real root in common;  
(3) Any real root of $f(x) + 3 = 0$ is greater than any real root of $f(x) - 1 = 0$;  
(4) Any real root of $f(x) + 5 = 0$ is less than any real root of $f(x) - 2 = 0$. The correct statements are.

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