
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 an integer  $n \ge 2$ , find all sustems of  $n$  functions $ f_1,..., f_n : R \to R$  such that for all  $x,y \in R$  $f_1(x)-f_2 (x)f_2(y)+ f_1(y) = 0$  $f_2(x^2)-f_3 (x)f_3(y)+ f_2(y^2) = 0$  $...$  $f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0$ 

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