
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 $f(x)=ax^{2}+b\ln x$ has an extreme value of $\frac{1}{2}$ at $x=1$.

(1) Find the values of $a$ and $b$;
(2) Find the monotonic intervals of the function $y=f(x)$;
(3) Find the extreme values of the function $f(x)$ on $x\in[\frac{1}{e},e]$ ($e$ is the base of the natural logarithm, $e=2.71828...$).

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