
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 rhombus ABCD,  $\angle A = 60^\circ$ . Rhombus  $BEFG$  is constructed, where  $E$  and  $G$  are the midpoints of  $BC$  and  $AB$ , respectively. Rhombus  $BHIJ$  is constructed, where  $H$  and  $J$  are the midpoints of  $BE$  and  $BG$ , respectively. This process is repeated forever. If the area of  $ABCD$  and the sum of the areas of all of the rhombi are both integers, compute the smallest possible value of  $AB$ .

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