
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  $n$  and  $k$  be integers,  $1\le k\le n$ . Find an integer  $b$  and a set  $A$  of  $n$  integers satisfying the following conditions:
(i) No product of  $k-1$  distinct elements of  $A$  is divisible by  $b$ .
(ii) Every product of  $k$  distinct elements of  $A$  is divisible by  $b$ .
(iii) For all distinct  $a,a'$  in  $A$ ,  $a$  does not divide  $a'$ .

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