
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  $\frac{1}{1}, \frac{1}{2}, ... , \frac{1}{2010}$  are written on a blackboard.  A student chooses any two of the numbers, say  $x$ ,  $y$ , erases them and then writes down  $x + y + xy$ . He continues to do this until only one number is left on the blackboard. What is this number?

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