
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 a positive integer \( n \geq 3 \), find the smallest integer \( k \) such that the following statement holds: For any \( n \) points \( A_i (x_i, y_i) \) in the plane, where no three points are collinear, and for any \( n \) real numbers \( c_i \), there always exists a real-coefficient bivariate polynomial \( P(x, y) \) of degree at most \( k \) satisfying \( P(x_i, y_i) = c_i \).

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