
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
The side length of a square is increased to $10$ units, each corner of the square contains a black isosceles right triangle with legs of length $3$ units, and a black diamond in the center has a side length of $3\sqrt{2}$. A circular coin with diameter $2$ units is dropped onto the square, landing such that the coin is entirely within the boundaries of the square. Find the probability that the coin will cover part of the black region on the square.
A) $\frac{77 + 5\pi + 11\sqrt{2}}{64}$ 
B) $\frac{78 + 5\pi + 12\sqrt{2}}{64}$ 
C) $\frac{79 + 6\pi + 13\sqrt{2}}{64}$ 
D) $\frac{80 + 4\pi + 14\sqrt{2}}{64}$

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