
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
Consider a coordinate plane where at each lattice point, there is a circle with radius $\frac{1}{8}$ and a square with sides of length $\frac{1}{4}$, whose sides are parallel to the coordinate axes. A line segment runs from $(0,0)$ to $(729, 243)$. Determine how many of these squares and how many of these circles are intersected by the line segment, and find the total count of intersections, i.e., $m + n$.

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