
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
Let the sequence $\{{a}_{n}\}:1,-2,-2,3,3,3,-4,-4,-4,-4,⋯,\underbrace{{(-1)}^{k-1}k,⋯,{(-1)}^{k-1}k}_{k个},⋯$ be defined such that when $\frac{(k-1)k}{2}＜n≤\frac{k(k+1)}{2}$ ($k∈{N}^{*}$), ${a}_{n}={(-1)}^{k-1}k$. Let ${S}_{n}={a}_{1}+{a}_{2}+⋯+{a}_{n}$ ($n∈{N}^{*}$).
1. Write out $S_{1}$, $S_{2}$, $S_{3}$, $S_{4}$.
2. Let ${b}_{k}={S}_{\frac{k(k+1)}{2}}$. Find the general formula for the sequence $\{b_{k}\}$.
3. For $l\in N^{*}$, define the set ${P}_{l}=\{n|\frac{{S}_{n}}{{a}_{n}}∈Z, n∈{N}^{*}, and 1≤n≤l\}$. Find the number of elements in the set $P_{2023}$.

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