
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
The famous German mathematician Dirichlet made significant achievements in the field of mathematics. The function named after him, $f(x) = \begin{cases} 1, & \text{if } x \text{ is rational} \\ 0, & \text{if } x \text{ is irrational} \end{cases}$, is called the Dirichlet function. Regarding the function $f(x)$, the following four propositions are given:  
① $f(f(x))=1$;  
② The function $f(x)$ is an even function;  
③ For any non-zero rational number $T$, $f(x+T)=f(x)$ holds true for any $x \in \mathbb{R}$;  
④ There exist three points $A(x_1, f(x_1))$, $B(x_2, f(x_2))$, and $C(x_3, f(x_3))$, such that $\triangle ABC$ is an equilateral triangle.  
Among these propositions, the true ones are numbered \_\_\_\_\_\_. (Write down the numbers of all correct propositions)

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