
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
Point $P$ moves along a vertical line that intersects side $AB$ of $\triangle PAB$. Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ respectively. As $P$ moves vertically, calculate how many of the four quantities listed change: (i) the length of the segment $MN$, (ii) the perimeter of $\triangle PAB$, (iii) the area of $\triangle PAB$, (iv) the area of trapezoid $ABNM$.

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