
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 $x$: $x^2 - x < 2nx$ ($n \in \mathbb{N}^*$), the number of integers in the solution set is denoted as $a_n$, and the sum of the first $n$ terms of the sequence $\{a_n\}$ is denoted as $S_n$. Find the value of $S_{100}$.

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