
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 that point $P$ moves perpendicularly to side $AB$ in $\triangle PAB$, $M$ and $N$ are the midpoints of sides $PA$ and $PB$ respectively, determine how many of the four quantities listed below change as $P$ is moved: the length of segment $MN$, the perimeter of the triangle $PAB$, the area of $\triangle PAB$, the area of trapezoid $ABNM$.

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