
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 $(C)$ with its center at the coordinate origin, foci on the $x$-axis, left vertex $A$, left focus $F_{1}(-2,0)$, and point $B(2, \sqrt{2})$ on the ellipse $(C)$. The line $y=kx (k \neq 0)$ intersects the ellipse $(C)$ at points $P$ and $Q$, and lines $AP$ and $AQ$ intersect the $y$-axis at points $M$ and $N$, respectively.

(1) Find the equation of the ellipse $(C)$.
(2) Find the coordinates of the fixed points through which the circle with $MN$ as its diameter passes.

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