
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
In the quadratic equation $x^{2}-2ax+b=0$, if $a^{2}-b \gt 0$, then $a$ is called the midpoint value of the equation.<br/>$(1)$ The midpoint value of the equation $x^{2}-8x+3=0$ is ______.<br/>$(2)$ Given that the midpoint value of the equation $x^{2}-mx+n=0$ is $3$, and one of the roots is exactly equal to $n$, find the value of $n$.

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