
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  $n_1,n_2,\ldots,n_s$  be distinct integers such that  $$ (n_1+k)(n_2+k)\cdots(n_s+k) $$  is an integral multiple of  $n_1n_2\cdots n_s$  for every integer  $k$ . For each of the following assertions give a proof or a counterexample: $(\text a)$   $|n_i|=1$  for some  $i$  $(\text b)$  If further all  $n_i$  are positive, then  $$ \{n_1,n_2,\ldots,n_2\}=\{1,2,\ldots,s\}. $$  

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