
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 $S$ be a square one of whose sides has a length of $\sqrt{2}$. A point $v=(x,y)$ is chosen uniformly at random over all pairs of real numbers $x$ and $y$ such that $0 \le x \le 5000$ and $0 \le y \le 5000$. Let $T(v)$ be a translated copy of $S$ centered at $v$. What is the probability that the square region determined by $T(v)$ contains exactly three points with integer coordinates in its border?
A) $\frac{1}{625}$
B) $\frac{2}{625}$
C) $\frac{3}{625}$
D) $\frac{4}{625}$
E) $\frac{5}{625}$

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