
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 ordered pairs \((2011,2),(2010,3),(2009,4), \ldots,(1008,1005),(1007,1006)\) are written from left to right on a blackboard. Every minute, Elizabeth selects a pair of adjacent pairs \(\left(x_{i}, y_{i}\right)\) and \(\left(x_{j}, y_{j}\right)\), with \(\left(x_{i}, y_{i}\right)\) to the left of \(\left(x_{j}, y_{j}\right)\), erases them, and writes \(\left(\frac{x_{i} y_{i} x_{j}}{y_{j}}, \frac{x_{i} y_{i} y_{j}}{x_{j}}\right)\) in their place. Elizabeth continues this process until only one ordered pair remains. How many possible ordered pairs \((x, y)\) could appear on the blackboard after the process has come to a conclusion?

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