
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
On the 6 by 6 square grid below, each dot is 1 cm from its nearest horizontal and vertical neighbors. Find the product of the area of square $PQRS$ (in cm$^2$) and the perimeter of square $PQRS$ (in cm) where the vertices $PQRS$ are positioned as follows on the grid: $P$ at (1,5), $Q$ at (5,6), $R$ at (6,2), and $S$ at (2,1). Express your answer in simplest radical form.

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