
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
Square  $ABCD$  has side length  $4$ . Points  $E$  and  $F$  are the midpoints of sides  $AB$  and  $CD$ , respectively. Eight  $1$  by  $2$  rectangles are placed inside the square so that no two of the eight rectangles overlap (see diagram). If the arrangement of eight rectangles is chosen randomly, then there are relatively prime positive integers  $m$  and  $n$  so that  $\tfrac{m}{n}$  is the probability that none of the rectangles crosses the line segment  $EF$  (as in the arrangement on the right). Find  $m + n$ .

[asy]
size(200);
defaultpen(linewidth(0.8)+fontsize(10pt));
real r = 7;
path square=origin--(4,0)--(4,4)--(0,4)--cycle;
draw(square^^shift((r,0))*square,linewidth(1));
draw((1,4)--(1,0)^^(3,4)--(3,0)^^(0,2)--(1,2)^^(1,3)--(3,3)^^(1,1)--(3,1)^^(2,3)--(2,1)^^(3,2)--(4,2));
draw(shift((r,0))*((2,4)--(2,0)^^(0,2)--(4,2)^^(0,1)--(4,1)^^(0,3)--(2,3)^^(3,4)--(3,2)));
label("A",(4,4),NE);
label("A",(4+r,4),NE);
label("B",(0,4),NW);
label("B",(r,4),NW);
label("C",(0,0),SW);
label("C",(r,0),SW);
label("D",(4,0),SE);
label("D",(4+r,0),SE);
label("E",(2,4),N);
label("E",(2+r,4),N);
label("F",(2,0),S);
label("F",(2+r,0),S);
[/asy]

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