
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 $C_1$ with its center at the origin, whose 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 $(-\sqrt{2}, 1)$. Let point $P$ be any point on ellipse $C_1$ and point $Q$ meet the conditions $\overrightarrow{AQ} \cdot \overrightarrow{AP} = 0$ and $\overrightarrow{BQ} \cdot \overrightarrow{BP} = 0$.
(1) Find the equation of ellipse $C_1$;
(2) Find the trajectory equation of point $Q$;
(3) When points $A$, $B$, and $Q$ are not collinear, find the maximum value of the area of triangle $ABQ$.

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