
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
Sheila is making a regular-hexagon-shaped sign with side length  $ 1$ . Let  $ABCDEF$  be the regular hexagon, and let  $R, S,T$  and U be the midpoints of  $FA$ ,  $BC$ ,  $CD$  and  $EF$ , respectively. Sheila splits the hexagon into four regions of equal width: trapezoids  $ABSR$ ,  $RSCF$  ,  $FCTU$ , and  $UTDE$ . She then paints the middle two regions gold. The fraction of the total hexagon that is gold can be written in the form  $m/n$  , where m and n are relatively prime positive integers. Compute  $m + n$ .
![Image](https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYS9lLzIwOTVmZmViZjU3OTMzZmRlMzFmMjM1ZWRmM2RkODMyMTA0ZjNlLnBuZw==&rn=MjAyMCBCTVQgSW5kaXZpZHVhbCAxMy5wbmc=)

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