
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
Suppose  $r \ge 2$  is an integer, and let  $m_1, n_1, m_2, n_2, \dots, m_r, n_r$  be  $2r$  integers such that   $$ \left|m_in_j-m_jn_i\right|=1 $$   for any two integers  $i$  and  $j$  satisfying  $1 \le i<j \le r$ . Determine the maximum possible value of  $r$ . 

*Proposed by B Sury*

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