
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 quadratic equation $ax^{2}+bx+c=0 (a\neq 0)$ satisfies $a+b+c=0$, then we call this equation a "Phoenix" equation.  
$(1)$ Write a "Phoenix" equation: ______;  
$(2)$ A "Phoenix" equation must have one root as: ______;  
$(3)$ Given the equation $x^{2}+mx+n=0$ is a "Phoenix" equation and has two equal real roots, find the value of $mn$.

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