
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 the set \( S = \{100, 101, 102, \ldots, 999, 1000\} \), and \( A \) be the set \( \{a_1, a_2, a_3, \ldots, a_n \mid a_1, a_2, \ldots, a_n \text{ are positive numbers, and } \frac{a_2}{a_1} = \frac{a_3}{a_2} = \cdots = \frac{a_n}{a_{n-1}} = q > 1 \} \). Find the maximum possible number of elements in the intersection \( S \cap A \).

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