
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 coordinates of the foci of a hyperbola $C$ are ${F_1}({-\sqrt{5},0})$ and ${F_2}({\sqrt{5},0})$, and the length of the real axis is $4$. <br/>$(1)$ Find the standard equation of the hyperbola $C$;<br/>$(2)$ If there exists a point $P$ on the hyperbola $C$ such that $PF_{1}\bot PF_{2}$, find the area of $\triangle PF_{1}F_{2}$.

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