
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 $f(x)=(x-a)^{2}(x+b)e^{x}$ where $a,b \in \mathbb{R}$.
(1) When $a=0$ and $b=-3$, find the monotonic intervals of the function $f(x)$.
(2) If $x=a$ is the maximum point of $f(x)$,
(i) When $a=0$, find the range of $b$;
(ii) When $a$ is a constant, suppose $x_{1}$, $x_{2}$, and $x_{3}$ (where $x_{1} < x_{2} < x_{3}$) are the three extreme points of $f(x)$. Does there exist a real number $b$ such that one can find a real number $x_{4}$ that makes $x_{4}$, $x_{1}$, $x_{2}$, and $x_{3}$ form an arithmetic sequence? If it exists, solve for the value of $b$ and the corresponding $x_{4}$. If not, explain the reason.

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