
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. What is the product of the value of the area of square $EFGH$ (in cm$^2$) and the value of the perimeter of square $EFGH$ (in cm)? Express your answer in simplest radical form.

[asy]
unitsize(1cm);
defaultpen(linewidth(0.7));
dot((0,0));
dot((0,1));
dot((0,2));
dot((0,3));
dot((0,4));
dot((0,5));
dot((1,0));
dot((1,1));
dot((1,2));
dot((1,3));
dot((1,4));
dot((1,5));
dot((2,0));
dot((2,1));
dot((2,2));
dot((2,3));
dot((2,4));
dot((2,5));
dot((3,0));
dot((3,1));
dot((3,2));
dot((3,3));
dot((3,4));
dot((3,5));
dot((4,0));
dot((4,1));
dot((4,2));
dot((4,3));
dot((4,4));
dot((4,5));
dot((5,0));
dot((5,1));
dot((5,2));
dot((5,3));
dot((5,4));
dot((5,5));
draw((1,5)--(5,5)--(5,1)--(1,1)--cycle);
label("$E$",(1,5),NW);
label("$F$",(5,5),NE);
label("$G$",(5,1),SE);
label("$H$",(1,1),SW);
[/asy]

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