
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
The domain of the function $f(x)$ is $(0,+\infty)$, and for all $x > 0$, $y > 0$, $f\left(\begin{matrix} \frac{x}{y}\end{matrix}\right)=f(x)-f(y)$. When $x > 1$, $f(x) > 0$.

(1) Find the value of $f(1)$;

(2) Determine the monotonicity of $f(x)$ and provide a proof;

(3) If $f(4)=2$, find the range of $f(x)$ on $[1,16]$.

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