
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 $(1/2, 3/2)$ and $(-1/2, -3/2)$. A point $v=(x,y)$ is chosen uniformly at random over all pairs of real numbers $x$ and $y$ such that $0 \le x \le 1006$ and $0\le y\le 1006$. 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) $\frac{1}{25}$
B) $\frac{2}{25}$
C) $\frac{1}{10}$
D) $\frac{1}{5}$
E) $\frac{3}{25}$

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