
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 functions $f(x) = (x+1)e^{x}$ and $g(x) = (e^{x} - a)(x - 1)^{2} (a > 0)$ where $e$ is the base of the natural logarithm.

(1) Find the monotonic intervals of the function $f(x)$;
(2) Determine the number of extreme points of the function $g(x)$ and explain the reason;
(3) If the function $g(x)$ has an extreme value of $2a^{2}$, find the value of $a$.

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