
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 left and right foci of an ellipse are $(-\sqrt{2}, 0)$ and $(\sqrt{2}, 0)$ respectively, point $A(\sqrt{2}, \frac{\sqrt{3}}{3})$ is on the ellipse $C$. The line $y=t$ intersects the ellipse $C$ at two distinct points $M$ and $N$. A circle $P$ is drawn with the diameter $MN$, and the center of the circle is denoted as $P$.  
(1) Find the equation of the ellipse $C$.  
(2) If the circle $P$ is tangent to the x-axis, find the coordinates of the center $P$.  
(3) Let $Q(x, y)$ be a moving point on the circle $P$. As $t$ varies, find the maximum value of $y$.

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