
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
In a new configuration, $EFGH$ is a square with side length $8,$ and $LMNO$ is a rectangle with $MO=12$ and $LO=8.$ Also, $EH$ and $LM$ are perpendicular. If the shaded area is equal to half of the area of $LMNO,$ what is the length of $EM?$

[asy]
draw((0,0)--(12,0)--(12,8)--(0,8)--cycle,black+linewidth(1));
draw((1.5,1)--(9.5,1)--(9.5,9)--(1.5,9)--cycle,black+linewidth(1));
filldraw((1.5,1)--(9.5,1)--(9.5,8)--(1.5,8)--cycle,gray,black+linewidth(1));
label("$L$",(0,8),NW);
label("$M$",(12,8),NE);
label("$O$",(12,0),SE);
label("$N$",(0,0),SW);
label("$E$",(1.5,9),NW);
label("$F$",(9.5,9),NE);
label("$G$",(9.5,1),E);
label("$H$",(1.5,1),SW);
label("$M$",(1.5,8),SW);
label("8",(1.5,9)--(9.5,9),N);
label("8",(12,0)--(12,8),E);
label("12",(0,0)--(12,0),S);
[/asy]

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