
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 $n$ be a given natural number. Find all continuous functions $f(x)$ such that
$$
(0) f(x) + (1) f\left(x^{2}\right) + (2) f\left(x^{2^{2}}\right) + \cdots + (n-1) f\left(x^{2^{n-1}}\right) + (n) f\left(x^{2^{n}}\right) = 0.
$$

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