
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
Assuming every lattice point in the coordinate plane has a circle of radius $\frac{1}{4}$ and a square of side length $\frac{1}{2}$ with sides parallel to the axes, determine how many squares and circles a line segment from $(0,0)$ to $(501, 201)$ intersects. Find the sum of these intersections.

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