
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
Let $R$ be a square, one diagonal of which has endpoints $(0,1)$ and $(1,0)$. A point $w=(x,y)$ is chosen uniformly at random such that $0 \le x \le 100$ and $0 \le y \le 100$. Let $Q(w)$ be a translated square centered at $w$. What is the probability that the region determined by $Q(w)$ contains exactly four points with integer coordinates on its perimeter?
A) $\frac{99}{100}$
B) $\frac{10000}{10201}$
C) $\frac{9801}{10000}$
D) $\frac{1}{2}$

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