
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 $PQRSTU$ be a regular hexagon. Let $V$, $W$, $X$, $Y$, $Z$, and $A$ be the midpoints of sides $PQ$, $QR$, $RS$, $ST$, $TU$, and $UP$, respectively. The segments $\overline{PV}$, $\overline{QW}$, $\overline{RX}$, $\overline{SY}$, $\overline{TZ}$, and $\overline{UA}$ bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of $PQRSTU$ 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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