
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 $F_{1}$ and $F_{2}$ are two foci of the ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$, with $|F_{1}F_{2}|=2$, and $M(2,\frac{2\sqrt{5}}{5})$ is a point on $C$. <br/>$(1)$ Find the standard equation of the ellipse $C$;<br/>$(2)$ If $P$ is a point on $C$ and $PF_{1}\bot PF_{2}$, find the area of $\triangle F_{1}PF_{2}$.

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