
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 plane rectangular coordinate system $xOy$, given a circle $O: x^{2}+y^{2}=1$, point $F(2,0)$, and a circle with diameter $FG$ tangent to circle $O$, where the center of the circle with diameter $FG$ is denoted as point $C$ and it is tangent to circle $O$ at point $H$. The locus of moving point $G$ is denoted as $W$.
- $(1)$ Find the equation of $W$.
- $(2)$ Let point $M$ be on the $x$-axis and point $N(0,1)$. Does there exist two points $A$ and $B$ on $W$ such that when $A$, $B$, and $N$ are collinear, $\triangle ABM$ is an isosceles right triangle with $AB$ as the hypotenuse? If yes, find the coordinates of point $M$ and the equation of line $AB$; if not, explain the reason.

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