
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
Determine the integers \( m \geq 2 \), \( n \geq 2 \), and \( k \geq 3 \) having the following property: \( m \) and \( n \) each have \( k \) positive divisors, and if \( d_1 < \cdots < d_k \) are the positive divisors of \( m \) (with \( d_1 = 1 \) and \( d_k = m \)) and \( d_1' < \cdots < d_k' \) are the positive divisors of \( n \) (with \( d_1' = 1 \) and \( d_k' = n \)), then \( d_i' = d_i + 1 \) for every integer \( i \) such that \( 2 \leq i \leq k - 1 \).

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