
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
Find the least positive integer \( n \) for which there exists a set \(\{s_{1}, s_{2}, \ldots, s_{n}\}\) consisting of \( n \) distinct positive integers such that
\[
\left(1-\frac{1}{s_{1}}\right)\left(1-\frac{1}{s_{2}}\right) \ldots\left(1-\frac{1}{s_{n}}\right)=\frac{51}{2010}.
\]

Same as above, but the constant \(\frac{51}{2010}\) is replaced by \(\frac{42}{2010}\).

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