
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 point P(0, 5) and circle C: $x^2+y^2+4x-12y+24=0$  
(1) Write down the coordinates of the center of circle C and its radius;  
(2) If line $l$ passes through point P and the segment intercepted by circle C has a length of $4\sqrt{3}$, find the equation of $l$;  
(3) Find the equation of the trajectory of the midpoint D of the chord of circle C that passes through point P.

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