
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
In the Cartesian coordinate system, two points are known: $A(-1,0)$ and $B(1,0)$. If the abscissa of a moving point $P(x,y)$ remains unchanged, and its ordinate is expanded to $\sqrt{2}$ times the original to get point $Q\left(x, \sqrt{2}y\right)$, and it satisfies $\vec{AQ} \cdot \vec{BQ}=1$.  
$(I)$ Find the equation of the curve $C$ on which point $P$ lies;  
$(II)$ A line $l$ passing through point $B$ with a slope of $- \frac{\sqrt{2}}{2}$ intersects curve $C$ at points $M$ and $N$, and $\vec{OM} + \vec{ON} + \vec{OH} = \vec{0}$. If point $H$ is symmetric to point $O$, and its symmetric point is point $G$, determine whether points $M$, $G$, $N$, $H$ are concyclic. If they are concyclic, find the coordinates of the center and the radius of the circle; if not, please explain why.

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