
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  $P(X,Y)=X^2+2aXY+Y^2$  be a real polynomial where  $|a|\geq 1$ . For a given positive integer  $n$ ,  $n\geq 2$  consider the system of equations:  \[ P(x_1,x_2) = P(x_2,x_3) = \ldots = P(x_{n-1},x_n) = P(x_n,x_1) = 0 . \] We call two solutions  $(x_1,x_2,\ldots,x_n)$  and  $(y_1,y_2,\ldots,y_n)$  of the system to be equivalent if there exists a real number  $\lambda \neq 0$ ,  $x_1=\lambda y_1$ ,  $\ldots$ ,  $x_n= \lambda y_n$ . How many nonequivalent solutions does the system have? 

*Mircea Becheanu*

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