
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 function $f(x) = a^x$ ($a > 0$ and $a \neq 1$),  
(1) If $f(x_0) = 2$, find the value of $f(3x_0)$.  
(2) If the graph of $f(x)$ passes through the point $(2, 4)$, let $g(x)$ denote the inverse function of $f(x)$. Determine the range of $g(x)$ on the interval $\left[ \frac{1}{2}, 2 \right]$.

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