
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));
for(int i = 0; i < 6; ++i){
    for(int j = 0; j < 6; ++j){
        dot((i, j));
    }
}
draw((1,4)--(4,5)--(5,2)--(2,1)--cycle);
label("$E$",(4,5),N);
label("$F$",(5,2),E);
label("$G$",(2,1),S);
label("$H$",(1,4),W);
[/asy]

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