
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
Let  $n$  be the answer to this problem. Suppose square  $ABCD$  has side-length  $3$ . Then, congruent non-overlapping squares  $EHGF$  and  $IHJK$  of side-length  $\frac{n}{6}$  are drawn such that  $A$ , $C$ , and  $H$  are collinear,  $E$  lies on  $BC$  and  $I$  lies on  $CD$ . Given that  $AJG$  is an equilateral triangle, then the area of  $AJG$  is  $a + b\sqrt{c}$ , where  $a$ ,  $b$ ,  $c$  are positive integers and  $c$  is not divisible by the square of any prime. Find  $a + b + c$ .


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