
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 the center C of circle C is in the first quadrant and lies on the line $3x-y=0$. The circle is tangent to the x-axis and the chord intercepted by the line $x-y=0$ has a length of $2\sqrt{7}$. The line $l: kx-y-2k+5=0$ intersects with circle C.  
(Ⅰ) Find the standard equation of circle C;  
(Ⅱ) Determine the fixed point through which line l passes; when the chord intercepted by the circle through line l is the shortest, find the equation of line l and the shortest length of the chord.

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