
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 $f(x)=ax^{2}-\ln x$ where $a\in R$.
(1) If the equation of the tangent line to the curve of $y=f(x)$ at the point $(1,f(1))$ is $y=x+b$ where $b\in R$, find the values of $a$ and $b$.
(2) If $y=f(x)$ takes an extreme value at $x=2$, find the maximum value of the function $f(x)$ in the interval $[\frac{1}{e}, e]$.

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