
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 functions $f(x) = \frac{1}{2}x^{2}$ and $g(x) = a\ln x (a > 0)$.
1. Find the extreme values of function $F(x) = f(x)g(x)$.
2. If function $G(x) = f(x) - g(x) + (a - 1)$ has two zero points in the interval $(\frac{1}{e}, e)$, find the range of $a$.
3. For function $h(x) = g(x) - x + \frac{1}{x}$, if $h(x\_2) - h(x\_1)$ has a maximum value $M(a)$ where $x\_1 \in (0,1)$ and $x\_2 \in (1, +\infty)$, does $M(a)$ have a maximum value when $a \leqslant e + \frac{1}{e}$? If it exists, find the maximum value; if not, explain the reason.

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