
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 two fixed circles ⊙F<sub>1</sub> with the equation $x^2 + y^2 + 4x + 3 = 0$ and ⊙F<sub>2</sub> with the equation $x^2 + y^2 - 4x - 5 = 0$, a moving circle M is tangent to both F<sub>1</sub> and F<sub>2</sub>, either externally or internally.  
(1) Find the equation of the locus C for the center M of the moving circle.  
(2) A line l through point F<sub>1</sub> intersects the locus C at points A and B, and intersects circle ⊙F<sub>2</sub> at points P and Q. If $|PQ| = 2$, find the length of $|AB|$.

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