
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
It is known that circle C is tangent to the y-axis, the center of the circle lies on the line $$x-3y=0$$, and the chord intercepted by the line $$y=x$$ has a length of $$2\sqrt{7}$$.  
(1) Find the equation of circle C.  
(2) If the center is in the first quadrant, find the equation of the line that passes through the point (6, 5) and is tangent to this circle.

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