
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 the parabola \( y^{2} = 4ax \) where \( 0 < a < 1 \) with a focus at \( F \). Taking \( A(a + 4, 0) \) as the center of a circle, and \(|AF|\) as the radius, draw a circle above the x-axis that intersects the parabola at two distinct points \( M \) and \( N \). Let \( P \) be the midpoint of \( MN \).

(1) Find the value of \( |MF| + |NF| \).

(2) Does a value of \( a \) exist such that \( |MF| \), \( |PF| \), and \( |NF| \) form an arithmetic sequence? If such a value exists, find \( a \); if it does not, provide an explanation.

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