
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
Triangle $ABC$ has $BC=24.$ The incircle of the triangle evenly quadrisects the median $AD$. If the area of the triangle is $p \sqrt{q}$ where $p$ and $q$ are integers and $q$ is not divisible by the square of a prime, find $p+q.$

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