
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 $P(-3,2)$ and circle $C: x^{2}+y^{2}+2x+4y+1=0$.  
$(1)$ Find the equation of the line passing through $P$ and tangent to circle $C$.  
$(2)$ Let circle $C$ intersect the circle with diameter $PC$ at points $A$ and $B$. Find $|AB|$.

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