
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 a circle of radius $48$, two chords of length $84$ intersect at a point whose distance from the center is $24$. Determine the area of the region that includes a smaller sector of the circle and one of the triangles formed by the chords and the intersection point. The area of this region can be expressed uniquely in the form $m\pi - n\sqrt{d}$, where $m, n,$ and $d$ are positive integers with $d$ not divisible by the square of any prime number. Find $m+n+d.$

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