
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
Given the function $f(x)= 
\begin{cases} 
1, & x \in \mathbb{Z}\\
0, & x \in \mathbb{R} \setminus \mathbb{Z}
\end{cases}$, where $\mathbb{Z}$ is the set of integers. Consider the following four propositions:

$① f(f(\sqrt{2}))=1$;

$② f(x)$ is an even function on $\mathbb{R}$;

$③$ For all $x_1, x_2 \in \mathbb{R}$, $f(x_1+x_2) \leqslant f(x_1) + f(x_2)$;

$④ f(x)$ is a periodic function with the smallest positive period being $1$.

Write down the number(s) of all correct propositions.

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