
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
A cylinder with a radius of $8$ units and a height of $10$ units is painted red. Two points, P and Q, are selected on the perimeter of the top base such that the arc PQ subtends an angle of $90^\text{o}$. The cylinder is cut along the plane passing through points P, Q, and the cylinder's axis. Determine the area of the newly exposed circular segment face using the form \( a \pi + b \sqrt{c} \), where \( a \), \( b \), and \( c \) are integers and \( c \) is square-free. Find the sum \( a+b+c \).

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