
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
Definition: If a one-variable quadratic equation about $x$, $ax^{2}+bx+c=0 (a\neq 0)$, satisfies $a-b+c=0$, then we call this equation a "golden equation".  
$(1)$ Determine whether the one-variable quadratic equation $4x^{2}+11x+7=0$ is a "golden equation", and explain the reason.  
$(2)$ Given that $3x^{2}-mx+n=0$ is a "golden equation" about $x$, if $m$ is a root of this equation, what is the value of $m$?

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