
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 circle $C:x^{2}+y^{2}-6x-4y+8=0$, a hyperbola is constructed such that one of its foci coincides with a point where the circle intersects the coordinate axes, and its vertex coincides with another intersection point. Determine the standard equation of the hyperbola that satisfies these conditions.

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