
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
A function $f(x)$ defined on $(0,+\infty)$ satisfies $\frac{{x}_{2}f({x}_{1})-{x}_{1}f({x}_{2})}{{x}_{1}-{x}_{2}}＜0$, and $f(\frac{1}{2})=3$, $f\left(3\right)=9$. The solution set of the inequality $f\left(x\right) \gt 3x$ is ______.

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