
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 that $a$ and $b$ are constants, $a \neq 0$, and $f(x)=ax^2+bx$, also we have $f(2)=0$. The equation $f(x)=x$ has two equal real roots.

(1) Find the explicit expression for $f(x)$.

(2) Does there exist real numbers $m$, $n$ $(m < n)$, such that the range of $f(x)$ on the interval $[m, n]$ is $[2m, 2n]$? If such $m$ and $n$ exist, find their values; if not, please explain why.

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