
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 $f(x)=ax^{2}+bx$ ($a,b$ are constants, and $a\neq 0$) satisfies the condition $f(-x+5)=f(x-3)$, and the equation $f(x)=x$ has two equal real roots.  
$(1)$ Find the expression of $f(x)$;  
$(2)$ Determine whether there exist real numbers $m$, $n$ ($m < n$), such that the domain and range of $f(x)$ are respectively $[m,n]$ and $[3m,3n]$, if they exist, find the values of $m$, $n$, if not, explain why.

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