
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
Suppose $r$, $s$, and $N$ are positive numbers with $s < 50$. Determine under what condition the number obtained by first increasing $N$ by $r\%$ and then decreasing it by $s\%$ exactly doubles the original number $N$.
**A)** $r = \frac{10000 + 100s}{100 + s}$   
**B)** $r = \frac{10000 + 100s}{100 - s}$   
**C)** $r = \frac{10000 - 100s}{100 + s}$  
**D)** $r = \frac{100s - 10000}{100 - s}$

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