
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 square in the coordinate plane has vertices at $(0, 0), (1000, 0), (1000, 1000),$ and $(0, 1000)$. Find the radius $d$ such that the probability that a randomly chosen point within the square is within $d$ units of a lattice point is $\tfrac{3}{4}$.
A) 0.4
B) 0.5
C) 0.6
D) 0.7

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