
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
The graph of the function \( f(x)=x+\frac{1}{x} \) resembles a checkmark, so we often refer to \( f(x) \) as the "checkmark function." Its graph is a hyperbola, and its asymptotes are the lines \( l_{1}: x=0 \) (i.e., the \( y \)-axis) and \( l_{2}: y=x \).

(1) Determine the coordinates of the vertex of the hyperbola and its eccentricity.
(2) Find the coordinates of the foci of the hyperbola.

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