
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 a quadratic function $f(x) = ax^2 + bx + c$ ($a \neq 0$) satisfies the conditions $f(-x+1) = f(x+1)$, $f(2) = 0$, and the equation $f(x) = x$ has two equal real roots.
(1) Find $a$, $b$, $c$.
(2) Determine whether there exist real numbers $m$, $n$ ($m < n$) such that the function $f(x)$ has a domain of $[m, n]$ and a range of $[3m, 3n]$. If such $m$ and $n$ exist, find their values; if not, explain why.

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