
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
Define a function $f(x)$ on $\mathbb{R}$ and its derivative $f'(x)$, both of whose graphs are continuous curves. Given real numbers $a, b$ ($a < b$) with $f'(a) > 0, f'(b) < 0$, consider the following statements:  
① There exists $x_0 \in [a,b]$ such that $f(x_0) = 0$;  
② There exists $x_0 \in [a,b]$ such that $f(x_0) > f(b)$;  
③ There exists $x_0 \in [a,b]$ such that $f(x_0) > f(a)$;  
④ There exists $x_0 \in [a,b]$ such that $f(a) - f(b) > f'(x_0)(a - b)$.  
Among these statements, the correct ones are.

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