
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 the ellipse $C: \frac{x^2}{2} + y^2 = 1$, where $F_1$ and $F_2$ are the left and right foci of the ellipse $C$, respectively.  
(I) Find the lengths of the major and minor axes of the ellipse $C$, the eccentricity $e$, and the left focus $F_1$;  
(II) Given that $P$ is a point on the ellipse and $PF_1 \perp PF_2$, find the area of $\triangle F_1PF_2$.

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