
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 the function $f(x)$ be an even function defined on $\mathbb{R}$, and for any $x \in \mathbb{R}$, it always holds that $f(x+1)=f(x-1)$. It is known that when $x \in [0,1]$, $f(x)=(\frac{1}{2})^{1-x}$. Then  
(1) $2$ is a period of the function $f(x)$;  
(2) The function $f(x)$ is decreasing on $(1,2)$ and increasing on $(2,3)$;  
(3) The maximum value of the function $f(x)$ is $1$, and the minimum value is $0$;  
(4) When $x \in (3,4)$, $f(x)=(\frac{1}{2})^{x-3}$.  
Among these statements, the correct ones are \_\_\_\_\_\_.

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