
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 a circle centered at the origin, the chord intercepted by the line $y=x+1$ has a length of $\sqrt{14}$.

(Ⅰ) Find the equation of the circle;

(Ⅱ) Let a variable line $y=k(x-1) \, (k \neq 0)$ intersect circle $C$ at points $A$ and $B$. Is there a fixed point $N$ on the positive half of the $x$-axis such that line $AN$ and line $BN$ are symmetric about the $x$-axis? If such a point exists, find the coordinates of point $N$; if not, explain why.

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