
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 the left and right foci of the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and point $Q(-\sqrt{2}, 1)$ lies on the ellipse. The line segment $QF_2$ intersects the $y$-axis at point $M$, and point $M$ is the midpoint of $QF_2$.

(1) Find the equation of the ellipse $C$;  
(2) Let $P$ be a point on the ellipse $C$, and $\angle F_1PF_2 = \frac{\pi}{2}$, find the area of $\triangle F_1PF_2$.

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