
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 that the graph of the power function $f(x)$ passes through the point $(3, \frac{1}{3})$.
$(1)$ Find the analytical expression of the function $f(x)$.
$(2)$ Let $g(x) = (x-2) \cdot f(x)$. Determine the monotonicity of the function $g(x)$ on the interval $[\frac{1}{2}, 1]$, and find the range of values of the function $g(x)$ on the interval $[\frac{1}{2}, 1]$.

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