
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
If the roots of the equation $x^{2}+px+q=0$ are $x_{1}$ and $x_{2}$, then $x_{1}+x_{2}=-p$ and $x_{1}\cdot x_{2}=q$. Use this conclusion to solve the following problems:
1. Given the equation $x^{2}+mx+n=0$ ($n\neq 0$), find a quadratic equation whose roots are the reciprocals of the roots of the given equation.
2. If $a$ and $b$ satisfy $a^{2}-15a-5=0$ and $b^{2}-15b-5=0$, find the value of $a+b$.
3. If $a$, $b$, and $c$ are real numbers such that $a+b+c=0$ and $abc=16$, find the minimum value of the positive number $c$.

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