
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  $m$ ,  $a$ ,  $b$ ,  $(a,b)=1$ .  $A$  is a non-empty subset of the set of all positive integers, so that for every positive integer  $n$  there is  $an \in A$  and  $bn \in A$ . For all  $A$  that satisfy the above condition, find the minimum of the value of  $\left| A \cap \{ 1,2, \cdots,m \} \right|$ 

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