
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
Equilateral triangle  $ABC$  has area  $1$ .  $A'$ ,  $B'$ , and  $C'$  are the midpoints of  $BC$ ,  $CA$ , and  $AB$ , respectively.  $A''$ ,  $B''$ ,  $C''$  are the midpoints of  $B'C'$ ,  $C'A'$ , and  $A'B'$ , respectively. The area of trapezoid  $BB''C''C$  can be written as  $\frac{m}{n}$  for relative prime positive integers  $m$  and  $n$ . Find  $m+n$ .

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