
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
Assume the coordinate plane now has a circle of radius $\frac{1}{8}$ and a square with side length $\frac{1}{4}$ centered at each lattice point, with the square’s sides parallel to the coordinate axes. Consider a diagonal line segment from $(0, 0)$ to $(401, 229)$ that intersects $m$ squares and $n$ circles. Determine the value of $m + n$.

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