
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
In the Cartesian coordinate system $(xOy)$, a circle $(O)$: $x^{2}+y^{2}=4$ intersects the negative semi-axis of $x$ at point $A$. Lines $AM$ and $AN$ pass through point $A$ and intersect circle $(O)$ at points $M$ and $N$ respectively.
1. If the slopes of lines $AM$ and $AN$ are $2$ and $-\frac{1}{2}$ respectively, find the area of triangle $AMN$.
2. A point $P(3\sqrt{3}, -5)$ is given. Two tangents are drawn from point $P$ to the circle $(O)$, intersecting the circle at points $E$ and $F$ respectively. Find the dot product of vectors $\overrightarrow{PE}$ and $\overrightarrow{PF}$.

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