
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 $ABCDEFGH$ be a regular octagon. Let $P$, $Q$, $R$, $S$, $T$, $U$, $V$, and $W$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, $FG$, $GH$, and $HA$, respectively. The segments $\overline{AP}$, $\overline{BQ}$, $\overline{CR}$, $\overline{DS}$, $\overline{ET}$, $\overline{FU}$, $\overline{GV}$, and $\overline{HW}$ bound a smaller regular octagon inside the original one. Determine the ratio of the area of the smaller octagon to the area of $ABCDEFGH$, expressed as a fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, and find $m+n$.

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