
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 a function $f(x)$ defined on $\mathbb{R}$ satisfies $f(x+y)=f(x)+f(y)$ for any real numbers $x, y$, and $f(x) > 0$ when $x > 0$. Consider the following statements:  
① $f(0)=1$;  
② $f(x)$ is an odd function on $\mathbb{R}$;  
③ $|f(x)|$ is an even function on $\mathbb{R}$;  
④ $f(x)$ is an increasing function on $\mathbb{R}$;  
⑤ $f(x)+1$ is a decreasing function on $\mathbb{R}$;  
Among these statements, the correct ones are.

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