
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 $M=\{f(x)\mid f^{2}(x)-f^{2}(y)=f(x+y)\cdot f(x-y), x,y\in \mathbb{R}\}$, determine the correctness of the following propositions:
1. If $f\_1(x)= \begin{cases}1,x\geqslant 0 \\ -1,x < 0\end{cases}$, then $f\_1(x)\in M$;
2. If $f\_2(x)=2x$, then $f\_2(x)\in M$;
3. If $f\_3(x)\in M$, then the graph of $y=f\_3(x)$ is symmetric about the origin;
4. If $f\_4(x)\in M$, then for any distinct real numbers $x\_1$, $x\_2$, we have $\frac{f\_4(x\_1)-f\_4(x\_2)}{x\_1-x\_2} < 0$.

The sequence numbers of all correct propositions are \_\_\_\_\_\_.

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