
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
Consider a larger square with a side length of $10$. Each corner of the square contains a black isosceles right triangle with legs of length $3$. In the very center of the square, there is a black diamond-shaped region with side length $3\sqrt{2}$. A circular coin with a diameter of $2$ is randomly dropped onto the square so that the coin is completely contained within the square. Calculate the probability that the coin will overlap with any black region. The answer can be expressed as $\frac{1}{k}\left(c+d\sqrt{2}+\pi\right)$, where $c$ and $d$ are integers. What is $c+d$?
A) 40
B) 41
C) 42
D) 43
E) 44

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