
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 $Q(- \sqrt{2}, 1)$ is on the ellipse. The line segment $QF\_2$ intersects the $y$-axis at point $M$, which is also the midpoint of $QF\_2$.
(1) Find the equation of ellipse $C$;
(2) Suppose $P$ is a point on the ellipse $C$, and $\angle F\_1 P F\_2 = \frac{\pi}{2}$, find the area of $\triangle F\_1 P F\_2$.

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