
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 $f(x)=a^{x}+log_{a}(x-1)$, where $a > 0$ and $a \neq 1$.

(1) If $a=\frac{1}{4}$, find the minimum value of $f(x)$ on the interval $x \in [1,2]$;

(2) If the minimum value of $f(x)$ on the interval $x \in [2,3]$ is 4, find the value of $a$.

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