
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
Centered at each lattice point in the coordinate plane are a circle of radius  $\tfrac{1}{10}$  and a square with sides of length  $\tfrac{1}{5}$  whose sides are parallel to the coordinate axes. The line segment from  $(0, 0)$  to  $(1001, 429)$  intersects  $m$  of the squares and  $n$  of the circles. Find  $m + n$ .

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