
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 side length $6$. Let $G$, $H$, $I$, $J$, $K$, and $L$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $AF$, respectively. Define points $P$, $Q$, $R$, $S$, and $T$ as the intersections of $\overline{AH}$ with $\overline{BI}$, $\overline{BI}$ with $\overline{CJ}$, $\overline{CJ}$ with $\overline{DK}$, $\overline{DK}$ with $\overline{EL}$, and $\overline{EL}$ with $\overline{FG}$ respectively, forming a pentagon. Calculate the ratio of the area of this pentagon to the area of hexagon $ABCDEF$ and express your answer in simplest fraction form.

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