
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 point is chosen at random within a square in the coordinate plane with vertices at $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$. Find the radius $d$ such that the probability a randomly selected point within the square is within $d$ units of a lattice point is $\tfrac{3}{4}$. (A lattice point $(x, y)$ has both $x$ and $y$ as integers.)
**A)** $0.3$  
**B)** $0.4$  
**C)** $0.5$  
**D)** $0.6$  
**E)** $0.7$

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