
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
Given is a natural number  $n \geq 3$ . What is the smallest possible value of  $k$  if the following statements are true?
For every  $n$  points  $ A_i = (x_i, y_i) $  on a plane, where no three points are collinear, and for any real numbers  $ c_i$   ( $1 \le i \le n$ ) there exists such polynomial  $P(x, y)$ , the degree of which is no more than  $k$ , where  $ P(x_i, y_i) = c_i $  for every  $i = 1, \dots, n$ .

(The degree of a nonzero monomial  $ a_{i,j} x^{i}y^{j} $  is  $i+j$ , while the degree of polynomial  $P(x, y)$  is the greatest degree of the degrees of its monomials.)

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