
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 that in $\triangle ABC$, $\tan A$ and $\tan B$ are the two real roots of the equation $x^{2} + ax + 4 = 0$:
1. If $a = -8$, find the value of $\tan C$;
2. Find the minimum value of $\tan C$, and specify the corresponding values of $\tan A$ and $\tan B$ in this case.

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