
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
Numbers  $1,\frac12,\frac13,\ldots,\frac1{2001}$  are written on a blackboard. A student erases two numbers  $x,y$  and writes down the number  $x+y+xy$  instead. Determine the number that will be written on the board after  $2000$  such operations.

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