
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 \(\frac{1}{1}, \frac{1}{2}, \ldots, \frac{1}{2010}\) 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}
user