
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
Four congruent isosceles triangles $DAO$, $AOB$, $BOC$, and $COD$ are arranged to consistently form a rhombus $ABCD$, with $DA = AO = OB = BC = CD = OD = 15$ and $AB = DC = 20$. Point $P$ is situated exactly at the midpoint of diagonal $AC$, with $O$ being the center of the rhombus.

Point $X$ is the midpoint of $ DA $ and point $Y$ is the midpoint of $BC$. When lines $XO$ and $OY$ are drawn, they divide the rhombus into four smaller triangular regions. Calculate the ratio of the area of triangle $ AXO $ to the area of triangle $ OYC $ in simplified form, and find $p+q$ if the ratio is expressed as $p:q$.

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