
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 that one asymptote of a hyperbola is $y-x=0$, and it passes through the point $(\sqrt{5}, 1)$.
(1) Find the standard equation of the hyperbola;
(2) If the line $y=kx-1$ intersects with the hyperbola obtained above at only one point, find the value of $k$.

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