
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 $ABCDEF$ be a regular hexagon with a side length of 4. Let $G$, $H$, $I$, $J$, $K$, and $L$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $AF$, respectively. A smaller regular hexagon is formed inside $ABCDEF$ by connecting points $M$, $N$, $O$, $P$, $Q$, $R$, which are the intersections of segments $\overline{AH}$, $\overline{BI}$, $\overline{CJ}$, $\overline{DK}$, $\overline{EL}$, and $\overline{FG}$, each scaled down by a factor of $\frac{1}{2}$ from the midpoint toward each vertex $A$, $B$, $C$, $D$, $E$, $F$ respectively. Find the ratio of the area of the smaller hexagon to the area of $ABCDEF$ and express it as a fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

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