
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 the logarithmic function $f(x) = \log_a x$ ($a>0$, $a \neq 1$) and the inverse proportion function $g(x) = \frac{k}{x}$, both graphs pass through the point $(2, \frac{1}{2})$.
(1) Find the expressions for $y=f(x)$ and $y=g(x)$;
(2) Solve the inequality $g[f(x)] < 2$ for $x$.

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