
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 set $M=\{f(x)|f^{2}(x)-f^{2}(y)=f(x+y)f(x-y),x,y∈R\}$, determine the correctness of the following propositions:

1. If $f(x)= \begin{cases} 1,x\geqslant 0 \\ -1,x < 0\end{cases}$, then $f(x)∈M$;
2. If $f(x)=2x$, then $f(x)∈M$;
3. If $f(x)∈M$, then the graph of $y=f(x)$ is symmetric about the origin;
4. If $f(x)∈M$, then for any real numbers $x_{1}$, $x_{2}(x_{1}≠ x_{2})$, we have $\frac {f(x_{1})-f(x_{2})}{x_{1}-x_{2}} < 0$.

Write out the sequence numbers of all correct propositions.

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