
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
In a square chessboard with dimensions $8 \times 8$ and a side length of 8, a circle with a radius of 4 is placed. Let $S_{1}$ denote the sum of the areas within the circle for all the small squares that the circle intersects, and $S_{2}$ denote the sum of the areas outside the circle for all the small squares that the circle intersects. Find the integer part of $\frac{S_{1}}{S_{2}}$.

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