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

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