
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 the function $f(x)=ax^{2}+bx$ ($a,b$ are constants, and $a\neq 0$) satisfies the condition $f(1-x)=f(1+x)$, and the function $g(x)=f(x)-x$ has only one zero point.

$(1)$ Find the expression of the function $f(x)$;

$(2)$ Find the real numbers $m$, $n$ ($m < n$), such that when the domain of $f(x)$ is $[m,n]$, the range of $f(x)$ is $[3m,3n]$.

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