
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 an ellipse C with foci at F<sub>1</sub> (-1, 0) and F<sub>2</sub> (1, 0), and the major axis length is $2\sqrt{5}$. The line $y=2x-2$ intersects the ellipse C at points A and B.  
(1) Find the standard equation of the ellipse;  
(2) Let O be the origin, find the area of $\triangle OAB$.

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