
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. Let $P$, $Q$, $R$, $S$, $T$, and $U$ be the points on sides $AB$, $BC$, $CD$, $DE$, $EF$, and $AF$, respectively, such that $AP = PB/2$, $BQ = QC/2$, $CR = RD/2$, $DS = SE/2$, $ET = TF/2$, and $FU = UA/2$. The segments $\overline{AR}$, $\overline{BQ}$, $\overline{CS}$, $\overline{DT}$, $\overline{EU}$, and $\overline{FP}$ bound a smaller regular hexagon inside $ABCDEF$. Let the ratio of the area of the smaller hexagon to the area of $ABCDEF$ be expressed 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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