
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 a sequence $A: a_{1}, a_{2}, \ldots, a_{n}, \ldots$, where $a_{1}=0$, $|a_{i+1}|=|a_{i}+1|$ for $i=1,2,\ldots,n,\ldots$. The sum of the first $n$ terms of sequence $A$ is denoted as $S_{n}$. $(Ⅰ)$ Find the value of $S_{3}$; $(Ⅱ)$ If $a_{5}=-2$, find the value of $S_{5}$; $(Ⅲ)$ Does there exist a sequence $A$ such that $S_{2022}=1011$? If it exists, find the value of $a_{2023}$; if not, explain the reason.

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