
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 trisects the median \(AD\). If \(A\) is a right angle and the area of the triangle is \(m \sqrt{n}\) where \(m\) and \(n\) are integers, and \(n\) is not divisible by the square of a prime, find \(m+n\).

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