
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
Given positive integers \( m, a, b \) where \( \gcd(a, b) = 1 \). Let \( A \) be a non-empty subset of the set of positive integers such that for any positive integer \( n \), \( an \in A \) or \( bn \in A \). Find the minimum value of \( |A \cap \{1, 2, \cdots, m\}| \) for all subsets \( A \) with the given properties.

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