
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 \((x_i, y_i)\) and \((x_j, y_j)\), with \((x_i, y_i)\) left of \((x_j, y_j)\), erases them, and writes \((\frac{x_i y_i x_j}{y_j}, \frac{x_i y_i y_j}{x_j})\) 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}
user