
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 $OBC$ have $AD=AO=OB=BC=15$ and $AB=DO=OC=20$. These triangles are arranged to form trapezoid $ABCD$, as shown. Point $P$ is on side $AB$ such that $OP$ is perpendicular to $AB$.

[asy]
pair A, B, C, D, O, P;
A= (8,10);
B=(22,10);
C=(30,0);
D=(0,0);
O=(15,0);
P=(15,10);
draw(A--B--C--D--A);
draw(A--O--B);
draw(O--P, dashed);
label("A", A, NW);
label("B", B, NE);
label("C", C, SE);
label("D", D, SW);
label("O", O, S);
label("P", P, N);
label("20", (D+O)/2, S);
label("20", (O+C)/2, S);
label("15", (A+D)/2, NW);
label("15", (B+C)/2, NE);
[/asy]

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. The ratio of the area of trapezoid $ABYX$ to the area of trapezoid $XYCD$ in simplified form is $p:q$. Find $p+q$.

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