
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 $(1/5,-1/5)$ and $(-1/5,1/5)$. A point $v=(x,y)$ is chosen uniformly at random over all pairs of real numbers $x$ and $y$ such that $-100 \le x \le 100$ and $-100 \le y \le 100$. Besides $T(v)$, a circle $C(v)$ with radius 1 and center at $v$ is also considered. What is the probability that the region determined by $T(v)$ and not overlapped by $C(v)$ contains exactly one point with integer coordinates in its interior?

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