
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 the square one of whose diagonals has endpoints $(1/8,5/8)$ and $(-1/8,-5/8)$. A point $v=(x,y)$ is chosen uniformly at random over all pairs of real numbers $x$ and $y$ such that $0 \le x \le 1000$ and $0 \le y \le 1000$. 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 two points with integer coordinates in its interior?
A) $\frac{1}{8}$
B) $\frac{7}{50}$
C) $\frac{4}{25}$
D) $\frac{1}{4}$
E) $\frac{8}{25}$

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