
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
Olya wrote fractions of the form \( \frac{1}{n} \) on cards, where \( n \) includes all possible divisors of \( 6^{100} \) (including one and the number itself). She then arranged these cards in some order. After that, she wrote on the board the number on the first card, then the sum of the numbers on the first and second cards, then the sum of the numbers on the first three cards, and so on, finally the sum of the numbers on all the cards. Each sum Olya wrote on the board as an irreducible fraction. What is the minimum number of different denominators that could appear on the board?

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