
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 a 6 by 6 square grid, 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));
for(int i = 0; i <= 5; ++i) {
    for(int j = 0; j <= 5; ++j) {
        dot((i,j));
    }
}
draw((1,5)--(5,5)--(5,1)--(1,1)--cycle);
label("$E$",(1,5),W);
label("$F$",(5,5),N);
label("$G$",(5,1),E);
label("$H$",(1,1),S);
[/asy]

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