
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
There are four statements as follows:  
(1) If $p \lor q$ is true and $p \land q$ is false, then $p$ and $q$ must be one true and one false;  
(2) If the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n = n^2 + n + 1$, where $n \in \mathbb{N}^*$, then $a_n = 2n$, where $n \in \mathbb{N}^*$;  
(3) If $f′(x) = 0$, then $f(x)$ has an extreme value at $x = x$;  
(4) If a function $f(x)$ defined on $\mathbb{R}$ satisfies $f(x + 2) = -f(x - 1)$, then $6$ is a period of the function $f(x)$.  
Among these four statements, the correct ones are numbered as follows.

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