
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 an ellipse $C: \frac{{x}^{2}}{16} + \frac{{y}^{2}}{4} = 1$, $P(x_{0}, y_{0})$ is a point outside the ellipse. Two tangent lines are drawn through $P$ to the ellipse $C$, with tangents $M$ and $N$ and intersection point $Q$ of line $MN$ and line $OP$. Points $A$ and $B$ are the two intersection points of line $OP$ and ellipse $C$.
1. Find the product of the slopes of lines $OP$ and $MN$.
2. Find the maximum area of triangle $AMN$.

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