
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 the equation $x^{2}+y^{2}-4x-2y-k=0$ represents a circle.
(1) If the circle symmetric to the line $x+y-4=0$ is tangent to the line $6x+8y-59=0$, find the value of the real number $k$;
(2) If $k=15$, find the equation of the smallest circle that passes through the intersection point of the given circle and the line $x-2y+5=0$.

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