
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. Construct square $ABCD$ whose vertices are positioned as follows: $A$ is at (5,5), $B$ is at (2,6), $C$ is at (1,3), and $D$ is at (4,2). Determine the product of the area of square $ABCD$ (in cm$^2$) and the perimeter of square $ABCD$ (in cm). Express your answer in simplest radical form. Assume each row and column contains 6 dots.

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