
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 $f(x)$ in its domain $(0,+\infty)$, $f(2)=1$, $f(xy)=f(x)+f(y)$; for $x>1$, $f(x)>0$:
(1) Find the value of $f(8)$.
(2) Discuss the monotonicity of $f(x)$ in its domain $(0,+\infty)$.
(3) Solve the inequality $f(x)+f(x-2)\leq 3$.

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