
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
In trapezoid $ABCD$, three congruent isosceles triangles $DAO$, $AOB$, and $OBC$ have $AD=AO=OB=BC=15$ and $AB=DO=OC=9$. These triangles are arranged with $AB$ and $CD$ as the non-parallel sides. Point $P$ lies on side $AB$ such that $OP$ is perpendicular to $AB$. Points $X$ and $Y$ are on sides $AD$ and $BC$, respectively, such that $AX=2XD$ and $BY=2YC$. Connect $X$ and $Y$. Calculate the ratio of the areas of trapezoids $ABYX$ to $XYCD$ and find the sum of these ratio numbers.

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