
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 \((\frac{1}{5}, \frac{8}{5})\) and \((- \frac{1}{5}, - \frac{8}{5})\). A point \(v = (x, y)\) is selected randomly such that \(x\) and \(y\) are real numbers with \(0 \leq x \leq 1000\) and \(0 \leq y \leq 1000\). Let \(T(v)\) be a translation of \(S\) centered at \(v\). What is the probability that the region defined by \(T(v)\) contains exactly three points with integer coordinates in its interior?
A) $\frac{1}{14}$
B) $\frac{7}{100}$
C) $\frac{1}{20}$
D) $\frac{1}{25}$

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