
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
A random point is chosen within a square in the coordinate plane with vertices at $(0, 0), (3030, 0), (3030, 3030),$ and $(0, 3030)$. Determine the side length $s$ of a square such that if a square of side length $s$ is centered at each lattice point inside the large square, the probability that the point lies inside at least one of these smaller squares is $\frac{1}{2}$.
A) $\frac{1}{2}$
B) $\frac{\sqrt{2}}{2}$
C) $\frac{\sqrt{3}}{2}$
D) 1

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