
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 \(y=f(x)\) be a decreasing function defined on \((0,+\infty)\) that satisfies \(f(xy)=f(x)+f(y)\), with \(f\left( \frac{1}{3} \right)=1\).

(1) Find the values of \(f(1)\), \(f\left( \frac{1}{9} \right)\), and \(f(9)\).

(2) If \(f(x)-f(2-x)<2\), determine the range of values for \(x\).

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