
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 equation $x^2 + px + q = 0$ with roots $x_1$ and $x_2$, we know that $x_1 + x_2 = -p$ and $x_1x_2 = q$. Using this information, solve the following problems:  
(1) Given the equation $x^2 + mx + n = 0$ ($n \ne 0$), find a quadratic equation whose roots are the reciprocals of the given equation's roots;  
(2) Given that $a$ and $b$ satisfy $a^2 - 15a - 5 = 0$ and $b^2 - 15b - 5 = 0$, find the value of $a + b$;  
(3) Given $a$, $b$, and $c$ are real numbers and $a + b + c = 0$ with $abc = 16$, find the minimum value of the positive number $c$.

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