
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
Six members of the team of Fatalia for the International Mathematical Olympiad are selected from  $13$  candidates. At the TST the candidates got  $a_1,a_2, \ldots, a_{13}$  points with  $a_i \neq a_j$  if  $i \neq j$ .

The team leader has already  $6$  candidates and now wants to see them and nobody other in the team. With that end in view he constructs a polynomial  $P(x)$  and finds the creative potential of each candidate by the formula  $c_i = P(a_i)$ .

For what minimum  $n$  can he always find a polynomial  $P(x)$  of degree not exceeding  $n$  such that the creative potential of all  $6$  candidates is strictly more than that of the  $7$  others?

*Proposed by F. Petrov, K. Sukhov*

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