
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 $\frac{\sqrt{2}}{2}$, and one of its foci is at $(\sqrt{2}, 0)$.

$(1)$ Find the equation of the ellipse $M$;  
$(2)$ Suppose a line $l$ intersects the ellipse $M$ at points $A$ and $B$, and 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 line $l$.

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