
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
Let the graph of the function $f(x)=x+ \frac{1}{x}$ ($x \in (-\infty, 0) \cup (0, +\infty)$) be $c_1$, and the symmetric graph of $c_1$ about point $A(2, 1)$ be $c_2$. The corresponding function for $c_2$ is denoted as $g(x)$.
(1) Find the explicit formula for the function $g(x)$ and determine its domain.
(2) If the line $y=b$ intersects $c_2$ at only one point, find the value of $b$ and the coordinates of the intersection point.

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