
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 $f(x) = \log_a(ax^2 - x + 1)$, where $a > 0$ and $a \neq 1$.
(1) When $a = \frac{1}{2}$, find the range of the function $f(x)$;
(2) When $f(x)$ is an increasing function on the interval $\left[\frac{1}{4}, \frac{3}{2}\right]$, find the range of the real number $a$.

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