
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 $ABC$, the medians $\overline{AD}$ and $\overline{CE}$ have lengths $15$ and $20$, respectively, and $AB=30$. Extend $\overline{CE}$ to intersect the circumcircle of $ABC$ at $G$. The area of triangle $AGB$ is $p\sqrt{q}$, where $p$ and $q$ are positive integers and $q$ is not divisible by the square of any prime. Find $p+q$.

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