
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 line segment $AB$ with endpoint $B(4,0)$, and endpoint $A$ moves on the circle $(x+4)^{2}+y^{2}=16$  
(Ⅰ) Find the trajectory equation of the midpoint $C$ of line segment $AB$.  
(Ⅱ) Suppose the moving line $y=k(x-1)$ ($k\neq 0$) intersects the circle $C$ at points $A$ and $B$. Is there a fixed point $N$ on the positive half of the $x$-axis such that the line $AN$ and the line $BN$ are symmetric about the $x$-axis? If it exists, find the coordinates of point $N$; if not, explain why.

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