
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
Point P is a moving point on circle A: $(x+1)^2+y^2=8$, and point B is $(1,0)$. The perpendicular bisector of segment PB intersects the radius PA at point M. Let the trajectory of point M be $\Gamma$.  
(I) Find the equation of curve $\Gamma$;  
(II) When point P is in the first quadrant, and $\cos\angle BAP= \frac{2\sqrt{2}}{3}$, find the coordinates of point M.

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