
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 M with its axes of symmetry being the coordinate axes, and its eccentricity is $\frac{\sqrt{2}}{2}$. The focus of the parabola $y^{2}=4\sqrt{2}x$ is also a focus of the ellipse M.  
(I) Find the equation of the ellipse M;  
(II) Suppose a line l intersects the ellipse M at points A and B. A parallelogram OAPB is formed with OA and OB as adjacent sides, where point P is on the ellipse M, and O is the origin. Find the minimum distance from point O to the line l.

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