
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  $\{\omega_1,\omega_2,\cdots,\omega_{100}\}$  be the roots of  $\frac{x^{101}-1}{x-1}$  (in some order). Consider the set   $$ S=\{\omega_1^1,\omega_2^2,\omega_3^3,\cdots,\omega_{100}^{100}\}. $$   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}
user