
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 center of ellipse $C_1$ is at the origin, and its foci are the vertices of the hyperbola $C_2: \frac{x^2}{2} - y^2 = 1$. The line $x + \sqrt{2}y = 0$ intersects the ellipse $C_1$ at points A and B, with the coordinates of point A being $(-\sqrt{2}, 1)$. Point P is any point on the ellipse $C_1$ other than points A and B, and point Q satisfies $\overrightarrow{AQ} \cdot \overrightarrow{AP} = 0$, $\overrightarrow{BQ} \cdot \overrightarrow{BP} = 0$, and points A, B, and Q are not collinear.
1. Find the equation of the ellipse $C_1$.
2. Find the equation of the trajectory of point Q.
3. Find the maximum area of $\triangle ABQ$ and the coordinates of point Q at this time.

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