
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 the function $f(x) = x^3 + bx^2 + cx + d$ (where $b$, $c$, and $d$ are constants), when $k \in (-\infty, 0) \cup (4, +\infty)$, the equation $f(x) - k = 0$ has only one real root; when $k \in (0, 4)$, the equation $f(x) - k = 0$ has three distinct real roots. Now, consider the following four statements:  
① The function $f(x)$ has two extreme points;  
② The function $f(x)$ has three extreme points;  
③ The equation $f(x) = 4$ and the equation $f'(x) = 0$ have one common real root;  
④ The equation $f(x) = 0$ and the equation $f'(x) = 0$ have one common real root;  
Among these, the correct statement(s) is/are.

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