
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 vertex coordinate of the quadratic function $f(x)$ is $(1,2)$, and its graph passes through the origin, the function $g(x)=\log_a x$ has its graph passing through the point $\left(\frac{1}{4}, -2\right)$.
(1) Find the analytical expressions for functions $f(x)$ and $g(x)$ respectively;
(2) Let the function $F(x) = g(f(x))$, determine the domain and range of $F(x)$.

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