
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 $f(x)$ be a function defined on $\mathbb{R}$ and not always equal to $0$, satisfying the following conditions: $\left(1\right) f(xy)=xf(y)+yf(x)$, $\left(2\right)$ when $x \gt 1$, $f(x) \lt 0$. Then which of the following conclusions is correct?  
① $f(1)=f(-1)=0$;  
② The function $f(x)$ is an odd function;  
③ The function $f(x)$ is decreasing on $(1,+\infty)$;  
④ The solution set of the inequality $\frac{{f(x)}}{x} \gt 0$ is $(-1,0) \cup (1,+\infty)$.

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