
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
Let 
\[ S_1=\{(x,y) \mid \log_{10}(1+x^2+y^2) \le \log_{10}(4(x+y)) \} \]
and 
\[ S_2=\{(x,y) \mid \log_{10}(1+x^2+y^2) \le \log_{10}(100(x+y)) \}. \]
What is the ratio of the area of \( S_2 \) to the area of \( S_1 \)?
A) 700
B) 714
C) 800
D) 1600

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