
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 large square in the coordinate plane has vertices at $(0, 0), (1500, 0), (1500, 1500),$ and $(0, 1500)$. A point is chosen at random within this square. Calculate the radius \( d \) such that the probability that the point is within \( d \) units of a lattice point is \( \frac{1}{3} \).
A) 0.2
B) 0.3
C) 0.4
D) 0.5
E) 0.6

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