
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 the function \( f(x) = x^2 - x + 1 \). Define \( f^{(n)}(x) \) as follows:
$$
f^{(1)}(x) = f(x), \quad f^{(n)}(x) = f\left(f^{(n-1)}(x)\right).
$$

Let \( r_{n} \) be the arithmetic mean of all the roots of \( f^{(n)}(x) = 0 \). Find \( r_{2015} \).

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