
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 $(a > 0, a \neq 1)$.

$(1)$ If $a= \frac{1}{2}$, find the range of the function $f(x)$.

$(2)$ When $f(x)$ is an increasing function in the interval $\left[ \frac{1}{4}, \frac{3}{2}\right]$, find the range of values for $a$.

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