
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 square with side length $10$ is colored white except for $4$ black isosceles right triangular regions with legs of length $3$ in each corner of the square, and a black diamond-shaped region with a side length $3\sqrt{2}$ at the center of the square. A circular coin with a diameter $2$ is dropped onto the square and lands in a random location where the coin is completely within the square borders. The probability that the coin will cover part of the black region can be written as $\frac{1}{100}\left(c+d\sqrt{2}+k\pi\right)$, where $c$, $d$, and $k$ are positive integers. What is $c+d+k$?
A) 60
B) 62
C) 64
D) 66
E) 68

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