
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 $ABCDEFGHIJKL$ be a regular dodecagon. Let $M$, $N$, $O$, $P$, $Q$, $R$, $S$, $T$, $U$, $V$, $W$, $X$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, $FG$, $GH$, $HI$, $IJ$, $JK$, $KL$, and $LA$ respectively. Segments $\overline{AM}$, $\overline{BN}$, $\overline{CO}$, $\overline{DP}$, $\overline{EQ}$, $\overline{FR}$, $\overline{GS}$, $\overline{HT}$, $\overline{IU}$, $\overline{JV}$, $\overline{KW}$, $\overline{LX}$ pass through a common center and are each perpendicular to their opposing side's segment. Let the ratio of the area of the inner dodecagon formed by these segments to the area of the larger dodecagon $ABCDEFGHIJKL$ 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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