
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
The numbers  $\frac{1}{1}, \frac{1}{2}, \cdots , \frac{1}{2012}$  are written on the blackboard. Aïcha chooses any two numbers from the blackboard, say  $x$  and  $y$ , erases them and she writes instead the number  $x + y + xy$ . She continues to do this until only one number is left on the board. What are the possible values of the final number?

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