
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: x² + y² + 4x - 12y + 24 = 0.
(1) Find the trajectory equation of the chord midpoints passing through point P in circle C.
(2) Point Q is a moving point on circle C. Find the trajectory equation of the midpoint M of PQ.

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