
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 $I$, $J$, $K$, $L$, $M$, $N$, $O$, and $P$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, $FG$, $GH$, and $HA$, respectively. The segments $\overline{AI}$, $\overline{BJ}$, $\overline{CK}$, $\overline{DL}$, $\overline{EM}$, $\overline{FN}$, $\overline{GO}$, and $\overline{HP}$ bound a smaller regular octagon inside the original. Find 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. Find $m + n$.

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