
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  $n\geq 2$  be an integer and let  $f$  be a  $4n$ -variable polynomial with real coefficients. Assume that, for any  $2n$  points  $(x_1,y_1),\dots,(x_{2n},y_{2n})$  in the Cartesian plane,  $f(x_1,y_1,\dots,x_{2n},y_{2n})=0$  if and only if the points form the vertices of a regular  $2n$ -gon in some order, or are all equal. 

Determine the smallest possible degree of  $f$ .

(Note, for example, that the degree of the polynomial   $$ g(x,y)=4x^3y^4+yx+x-2 $$   is  $7$  because  $7=3+4$ .)

*Ankan Bhattacharya*

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