
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
Let $P$ be a point on the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and $F_{1}$, $F_{2}$ be the foci of $C$. It is given that $PF_{1} \perp PF_{2}$.  
$(1)$ If $|PF_{1}| = 2|PF_{2}|$, find the eccentricity of the ellipse $C$.  
$(2)$ If the coordinates of point $P$ are $(3,4)$, find the standard equation of the ellipse $C$.

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