
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 the function $f(x)$ ($x \in \mathbb{N}$) represent the remainder when $x$ is divided by 2, and the function $g(x)$ ($x \in \mathbb{N}$) represent the remainder when $x$ is divided by 3. For any $x \in \mathbb{N}$, consider the following statements:  
① $f(x) \neq g(x)$; ② $g(2x) = 2g(x)$; ③ $f(2x) = 0$; ④ $f(x) + f(x+3) = 1$. Among these, the correct statement(s) is/are. (Write down all the statement numbers that apply)

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