
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
Suppose \( P \) is a point moving along a circular path centered at midpoint \( C \) of side \( AB \) in triangle \( \triangle PAB \). Points \( M \) and \( N \) remain defined as midpoints of sides \( PA \) and \( PB \) respectively. As \( P \) moves on this path, how many of the four quantities listed below change?
(a) the length of segment \( MN \)
(b) the perimeter of \( \triangle PAB \)
(c) the area of \( \triangle PAB \)
(d) the area of trapezoid \( ABMN \)
A) 0
B) 1
C) 2
D) 3
E) 4

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