
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 ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{2} = 1$, $A$, $B$ are its left and right vertices, and the moving point $M$ satisfies $MB \perp AB$. Connect $AM$ intersects the ellipse at point $P$, and there is a fixed point $Q$ on the $x$-axis distinct from points $A$, $B$. The circle with $MP$ as the diameter passes through the intersection of line $BP$ and line $MQ$. Find the coordinates of point $Q$.

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