
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 with radius $50$, two chords of length $84$ intersect at a point that is $24$ units away from the center of the circle. The intersection divides the interior of the circle into four regions. Calculate the area of one of the regions that is bordered by segments of unequal lengths. Express the area in the form $m\pi - n\sqrt{d}$, where $m, n,$ and $d$ are integers with $d$ not divisible by the square of any prime number, and find $m+n+d$.

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