
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
If the domain of $y=f(x)$ is $\mathbb{R}$, and for any $x$ in the domain, there exists a real number $a$ such that $f(x+a)=f(-x)$ holds, then this function is said to have the "P(a) property". Consider the following statements:  
1. The function $y=\sin x$ has the "P(a) property";  
2. If an odd function $y=f(x)$ has the "P(2) property", and $f(1)=1$, then $f(2015)=1$;  
3. If a function $y=f(x)$ has the "P(4) property", its graph is centrally symmetric about the point $(1,0)$, and it is monotonically decreasing at $(-1,0)$, then $y=f(x)$ is monotonically decreasing in $(-2,-1)$ and monotonically increasing in $(1,2)$;  
4. If a non-zero function $y=f(x)$ has both the "P(0) property" and the "P(3) property", and for any $x_{1}$, $x_{2} \in \mathbb{R}$, $|f(x_{1})-f(x_{2})| \geqslant |g(x_{1})-g(x_{2})|$ holds for the function $y=g(x)$, then the function $y=g(x)$ is a periodic function.  
The correct statements are __________ (Write down the numbers of all correct statements).

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