
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 rectangle of vertices at (0, 0), (2020, 0), (2020, 4040), and (0, 4040) has lattice points at each coordinate pair (x, y) where both x and y are integers. A randomly chosen point within the rectangle is picked. what is the radius \(d\) such that the probability that this point is within \(d\) units of any lattice point is \(\frac{1}{2}\)?
A) 0.2
B) 0.3
C) 0.4
D) 0.5
E) 0.6

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