
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 \(\omega_{1}, \omega_{2}, \ldots, \omega_{100}\) be the roots of \(\frac{x^{101}-1}{x-1}\) (in some order). Consider the set
\[
S=\left\{\omega_{1}^{1}, \omega_{2}^{2}, \omega_{3}^{3}, \ldots, \omega_{100}^{100}\right\}.
\]
Let \(M\) be the maximum possible number of unique values in \(S\), and let \(N\) be the minimum possible number of unique values in \(S\). Find \(M-N\).

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