
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 that the extreme point of the function $f(x) = x^{2} + a\ln x$ is $2$.
(1) Find the value of the real number $a$.
(2) Find the extreme value of the function $f(x)$.
(3) Find the maximum and minimum values of the function $f(x)$ on the interval $[\frac{1}{e}, e]$.

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