
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
Let \( x_1 = y_1 = x_2 = y_2 = 1 \). For \( n \geq 3 \), define sequences by \( x_n = x_{n-1}y_{n-2} + x_{n-2}y_{n-1} \) and \( y_n = y_{n-1}y_{n-2} - x_{n-1}x_{n-2} \). What are the last two digits of \( |x_{2012}| \)?

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