
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 $f(x)$ be an even function defined on $\mathbb{R}$ such that $f(x+1)=f(x-1)$ for any $x\in\mathbb{R}$. Given that $f(x)=2^{x}$ for $x\in[0,1]$, determine which of the following statements are true:

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 $0$.

Identify the sequence number(s) of all correct propositions.

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