
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
For what positive integers  $n$  and  $k$  do there exits integers  $a_1,a_2,\dots,a_n$  and  $b_1,b_2,\dots,b_k$  such that the products  $a_ib_j$  ( $1\le i\le n,1\le j\le k$ ) give pairwise different residues modulo  $nk$ ?

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