
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
The diagram below shows the regular hexagon  $BCEGHJ$  surrounded by the rectangle  $ADFI$ . Let  $\theta$  be the measure of the acute angle between the side  $\overline{EG}$  of the hexagon and the diagonal of the rectangle  $\overline{AF}$ . There are relatively prime positive integers  $m$  and  $n$  so that  $\sin^2\theta  = \tfrac{m}{n}$ . Find  $m + n$ .

[asy]
import graph; size(3.2cm); 
real labelscalefactor = 0.5; 
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
draw((-1,3)--(-1,2)--(-0.13,1.5)--(0.73,2)--(0.73,3)--(-0.13,3.5)--cycle); 
draw((-1,3)--(-1,2)); 
draw((-1,2)--(-0.13,1.5)); 
draw((-0.13,1.5)--(0.73,2)); 
draw((0.73,2)--(0.73,3)); 
draw((0.73,3)--(-0.13,3.5)); 
draw((-0.13,3.5)--(-1,3)); 
draw((-1,3.5)--(0.73,3.5)); 
draw((0.73,3.5)--(0.73,1.5)); 
draw((-1,1.5)--(0.73,1.5)); 
draw((-1,3.5)--(-1,1.5)); 
label(" $ A $ ",(-1.4,3.9),SE*labelscalefactor); 
label(" $ B $ ",(-1.4,3.28),SE*labelscalefactor); 
label(" $ C $ ",(-1.4,2.29),SE*labelscalefactor); 
label(" $ D $ ",(-1.4,1.45),SE*labelscalefactor); 
label(" $ E $ ",(-0.3,1.4),SE*labelscalefactor); 
label(" $ F $ ",(0.8,1.45),SE*labelscalefactor); 
label(" $ G $ ",(0.8,2.24),SE*labelscalefactor); 
label(" $ H $ ",(0.8,3.26),SE*labelscalefactor); 
label(" $ I $ ",(0.8,3.9),SE*labelscalefactor); 
label(" $ J $ ",(-0.25,3.9),SE*labelscalefactor); [/asy]

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