Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\triangle PAB$. As $P$ moves along a line that is parallel to side $AB$, find how many of the following four quantities change:  
  
$(a)$ the length of the segment $MN$  
  
$(b)$ the perimeter of $\triangle PAB$  
  
$(c)$ the area of $\triangle PAB$  
  
$(d)$ the area of trapezoid $ABNM$

Let's think step by step and output the final answer within \boxed{}.
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