
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
Given that $S$ is a square one of whose diagonals has endpoints $(1/8,3/8)$ and $(-1/8,-3/8)$, and $T(v)$ is a translated copy of $S$ centered at $v$, where $v=(x,y)$ and $0 \le x \le 1000$ and $0 \le y \le 1000$, find the probability that the square region determined by $T(v)$ contains exactly three points with integer coordinates in its interior.

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