
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
We wrote the reciprocals of natural numbers from 2 to 2011 on a board. In one step, we erase two numbers, \( x \) and \( y \), and replace them with the number

$$
\frac{xy}{xy + (1 - x)(1 - y)}
$$

By repeating this process 2009 times, only one number remains. What could this number be?

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