
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 \( k \geq 3 \) that satisfy the following properties: both \( m \) and \( n \) have exactly \( k \) positive divisors, and if \( d_{1}<\ldots<d_{k} \) are the positive divisors of \( m \) (with \( d_{1}=1 \) and \( d_{k}=m \)) and \( d_{1}^{\prime}<\ldots<d_{k}^{\prime} \) are the positive divisors of \( n \) (with \( d_{1}^{\prime}=1 \) and \( d_{k}^{\prime}=n \)), then \( d_{i}^{\prime}=d_{i}+1 \) for every integer \( i \) such that \( 2 \leq i \leq k-1 \).

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