
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 the function $y=f(x)$ be defined on the positive real numbers $R_+$, satisfying the following three conditions: (1) For any positive numbers $x$, $y$, it holds that $f(xy) = f(x) + f(y)$; (2) For $x > 1$, $f(x) < 0$; and (3) $f(3) = -1$.  
(a) Determine the values of $f(1)$ and $f\left(\frac{1}{9}\right)$;  
(b) Determine the monotonicity of the function and provide a proof;  
(c) If the inequality $f(x) + f(2-x) < 2$ holds, find the range of values for $x$.

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