
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 \( n > 1 \), find all real-valued functions \( f_i(x) \) on the reals such that for all \( x, y \) we have:
\[
f_1(x) + f_1(y) = f_2(x) f_2(y)
\]
\[
f_2(x^2) + f_2(y^2) = f_3(x) f_3(y)
\]
\[
f_3(x^3) + f_3(y^3) = f_4(x) f_4(y)
\]
\[
\vdots
\]
\[
f_n(x^n) + f_n(y^n) = f_1(x) f_1(y).
\]

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