
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
Circles $M$ and $P$ with the equation $x^2+y^2-2\sqrt{2}x-10=0$ are internally tangent, and both pass through the fixed point $Q(-\sqrt{2}, 0)$.
(I) Find the equation of the locus of the center of the moving circle $M$.
(II) A line $l$ with slope $\sqrt{3}$ intersects the locus of the center $M$ at points $A$ and $B$, and the perpendicular bisector of the segment $AB$ passes through the point $(0, -\frac{1}{2})$. Find the equation of line $l$.

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