
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  $r$  be a positive integer. Find the smallest positive integer  $m$  satisfying the condition: For all sets  $A_1, A_2, \dots, A_r$  with  $A_i \cap A_j = \emptyset$ , for all  $i \neq j$ , and  $\bigcup_{k = 1}^{r} A_k = \{ 1, 2, \dots, m \}$ , there exists  $a, b \in A_k$  for some  $k$  such that  $1 \leq \frac{b}{a} \leq 1 + \frac{1}{2022}$ .

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