
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 cylindrical block with a radius of $5$ and a height of $10$ has been entirely painted red. Two points, $A$ and $B$, are marked on the top circular edge such that the arc $\overarc{AB}$ spans an angle of $90^\text{o}$. The cylinder is cut through the plane that passes through points $A$, $B$, and the cylinder’s center, revealing two new flat, unpainted surfaces. Calculate the total area of one of these unpainted faces expressed in the form $a\pi + b\sqrt{c}$, where $a$, $b$, and $c$ are integers, with $c$ not divisible by the square of any prime. Find $a+b+c$.

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