
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
For a function $f(x)$ defined on the interval $[a,b]$, the following statements are given:  
$(1)$ If $f(x)$ attains local maximum values at several points, then the maximum value of $f(x)$ must be the largest one among all local maximum values;  
$(2)$ If the local maximum value of the function $f(x)$ is $m$, and the local minimum value is $n$, then $m > n$;  
$(3)$ If $x_{0} \in (a,b)$, and near the left side of $x_{0}$, $f'(x) < 0$, and $f'(x_{0})=0$, then $x_{0}$ is a local maximum point of $f(x)$;  
$(4)$ If $f'(x)$ is always positive on $[a,b]$, then $f(x)$ is an increasing function on $[a,b]$,  
Among these, the correct statement(s) is/are __________.

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