
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 a function $f(x)$ defined on the interval $(0,+\infty)$ that satisfies $f\left( \frac{x_{1}}{x_{2}} \right)=f(x_{1})-f(x_{2})$, and when $x > 1$, $f(x) > 0$, $f(3)=1$.

$(1)$ Determine the monotonicity of $f(x)$;

$(2)$ Solve the inequality $f(3x+6)+f\left( \frac{1}{x} \right) > 2$ with respect to $x$;

$(3)$ If $f(x)\leqslant m^{2}-2am+1$ holds for all $x\in(0,3]$ and $a\in[-1,1]$, find the range of the real number $m$.

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