
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 \( S \) be a set of positive integers satisfying the following two conditions:
- For each positive integer \( n \), at least one of \( n, 2n, \ldots, 100n \) is in \( S \).
- If \( a_1, a_2, b_1, b_2 \) are positive integers such that \( \operatorname{gcd}(a_1 a_2, b_1 b_2) = 1 \) and \( a_1 b_1, a_2 b_2 \in S \), then \( a_2 b_1, a_1 b_2 \in S \).

Suppose that \( S \) has natural density \( r \). Compute the minimum possible value of \( \left\lfloor 10^5 r \right\rfloor \).

Note: \( S \) has natural density \( r \) if \( \frac{1}{n}|S \cap \{1, \ldots, n\}| \) approaches \( r \) as \( n \) approaches \(\infty\).

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