
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 triangle $XYZ$, the medians $\overline{XM}$ and $\overline{YN}$ have lengths $15$ and $20$, respectively, and $XY = 20$. Extend median $\overline{YN}$ to intersect the circumcircle of $XYZ$ at $P$. The area of triangle $XPY$ is $a\sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime. Find $a+b$.

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