
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
The focus of the parabola \( y^2 = 4ax \) (where \( a > 0 \)) is \( A \). Consider a semicircle centered at \( B(a+4, 0) \) with radius \( |AB| \), positioned above the \( x \)-axis. Assume the semicircle intersects the parabola at distinct points \( M \) and \( N \), with \( P \) being the midpoint of the segment \( MN \). 

(I) Find the value of \( |AM| + |AN| \). 
(II) Does there exist a value of \( a \) such that \( |AM|, |AP|, and |AN| \) form an arithmetic sequence?

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