
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 the circle $O: x^{2}+y^{2}=16$, point $A(6,0)$, point $B$ is a moving point on circle $O$, and the locus of the midpoint $M$ of line segment $AB$ is curve $C$.
$(1)$ Find the equation of curve $C$;
$(2)$ Let $T(2,0)$, a line $l$ not coinciding with the $x$-axis passes through point $T$ and intersects curve $C$ at points $E$ and $F$.
$(i)$ A line $m$ perpendicular to line $l$ passes through point $T$ and intersects curve $C$ at points $G$ and $H$. Find the maximum area of quadrilateral $EGFH$;
$(ii)$ Let curve $C$ intersect the $x$-axis at points $P$ and $Q$. Line $PE$ intersects line $QF$ at point $N$. Discuss whether point $N$ lies on a fixed line. If yes, find the equation of that line; if not, explain the reason.

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