
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
We call a number  $n$  *interesting*if for each permutation  $\sigma$  of  $1,2,\ldots,n$  there exist polynomials  $P_1,P_2,\ldots ,P_n$  and  $\epsilon > 0$  such that: $i)$   $P_1(0)=P_2(0)=\ldots =P_n(0)$  $ii)$   $P_1(x)>P_2(x)>\ldots >P_n(x)$  for  $-\epsilon<x<0$  $iii)$   $P_{\sigma (1)} (x)>P_{\sigma (2)}(x)> \ldots >P_{\sigma (n)} (x) $  for  $0<x<\epsilon$ Find all *interesting* $n$ .

*Proposed by Mojtaba Zare Bidaki*

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