
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
The point $P$ moves circumferentially around the square $ABCD$. Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ respectively. Considering this motion, how many of the four quantities listed change: the length of the diagonal $AC$, the perimeter of square $ABCD$, the area of square $ABCD$, the area of quadrilateral $MBCN$?

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