
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 $f(x)=x+\sin 2x$. Consider the following four statements:
1. For all $x > 0$, the inequality $f(x) < 2x$ always holds;
2. There exists a $k \in \mathbb{R}$ such that the equation $f(x)=k$ has four distinct real roots;
3. The graph of the function $f(x)$ has infinitely many centers of symmetry;
4. If the sequence $\{a_n\}$ is an arithmetic sequence, and $f(a_1) + f(a_2) + f(a_3) = 3\pi$, then $a_2=\pi$.

Indicate all the correct statements by listing their numbers.

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