
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. Calculate the product of the area of square $EFGH$ and the perimeter of square $EFGH$, where square $EFGH$ is defined by the vertices $E(1,5)$, $F(5,6)$, $G(6,2)$, and $H(2,1)$. 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((1,5)--(5,6)--(6,2)--(2,1)--cycle);
label("$E$",(1,5),NW);
label("$F$",(5,6),N);
label("$G$",(6,2),E);
label("$H$",(2,1),S);
[/asy]

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