
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. Find the equation of line $ι$ that passes through $P$ and the length of the chord intercepted by circle $C$ is $4 \sqrt {3}$.
2. Find the trajectory equation of the midpoints of the chords of circle $C$ that pass through point $P$.

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