
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
Three congruent isosceles triangles $DAO$, $AOB$, and $OBC$ are arranged to form a trapezoid $ABCD$ with $AD=AO=OB=BC=13$ and $AB=DO=OC=15$. Point $P$ is on side $AB$ such that $OP$ is perpendicular to $AB$. Point $X$ is the midpoint of $AD$, and point $Y$ is the midpoint of $BC$. When $X$ and $Y$ are joined, the trapezoid is divided into two smaller trapezoids. Determine the ratio of the area of trapezoid $ABYX$ to the area of trapezoid $XYCD$ in simplified form and find the sum $p+q$ where the ratio is $p:q$.

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