
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 constants $a$ and $b$, with $a \neq 0$, and $f(x) = ax^2 + bx$, also $f(2) = 0$, and the equation $f(x) = x$ has two equal real roots.  
(1) Find the expression for $f(x)$.  
(2) Does there exist $m, n$ ($m < n$) such that the range of $f(x)$ in the interval $[m, n]$ is $[2m, 2n]$? If it exists, find the values of $m$ and $n$; if not, explain why.

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