
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 each side length $s$. Let $P$, $Q$, $R$, $S$, $T$, and $U$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $FA$, respectively. The segments $\overline{AP}$, $\overline{BQ}$, $\overline{CR}$, $\overline{DS}$, $\overline{ET}$, and $\overline{FU}$ bound a smaller regular hexagon inside $ABCDEF$. Determine the ratio of the area of this smaller hexagon to the area of $ABCDEF$, and express this ratio in its simplest form.

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