
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]


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Given an ellipse centered at the origin with the axes of symmetry being the coordinate axes, the focus of the parabola $y^{2}=16x$ as one of its foci, and the focus of the hyperbola $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$ as the vertex.

(1) Find the standard equation of the ellipse;

(2) Let $E$ and $F$ be two points on the ellipse that are symmetric about the origin, and let $P$ be a point on the ellipse. When the slopes of the lines $PE$ and $PF$ both exist and are denoted as $k_{PE}$ and $k_{PF}$, is $k_{PE}\cdot k_{PF}$ a constant value? If it is, find this constant value; otherwise, explain the reason.

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