
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 inequality about $x$: $x^{2}-x < 2nx (n \in \mathbb{N}^{*})$, let $a_{n}$ denote the number of integers in the solution set of the inequality. If $S_{n}$ represents the sum of the first $n$ terms of the sequence ${a_{n}}$, find the value of $S_{100}=\_\_\_\_\_\_.$

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