
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 any real number $x$, $\lfloor x \rfloor$ represents the largest integer not greater than $x$, such as $\lfloor 3.6 \rfloor = 3$, $\lfloor -3.6 \rfloor = -4$. For the function $f(x) = \left \lfloor \frac{x+1}{3} - \left \lfloor \frac{x}{3} \right \rfloor \right \rfloor$, the following statements are given:

1. $f(x)$ is a periodic function;
2. $f(x)$ is an even function;
3. The range of the function $f(x)$ is $\{0, 1\}$;
4. The function $g(x) = f(x) - \cos \pi x$ has two different zero points in the interval $(0, \pi)$,

The correct statement(s) is/are \_\_\_\_\_\_ (Fill in the blank with the number of the correct answer).

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