
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 the foci of an ellipse are $F_{1}(-4,0)$ and $F_{2}(4,0)$, and the eccentricity is $e=0.8$.  
$(1)$ Find the standard equation of the ellipse.  
$(2)$ Determine if there exists a point $P$ on the ellipse such that $\overrightarrow{{F}_{1}P}\cdot \overrightarrow{{F}_{2}P}=0$. If it exists, find the coordinates of $P$.

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