
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
What is \( x_{2010} \) based on the recurrence relation \( x_{k} = x_{k-1} + 1 - 4 \left( \left\lfloor \frac{k-1}{4} \right\rfloor - \left\lfloor \frac{k-2}{4} \right\rfloor \right) \) for \( k \geqslant 2 \) and initial condition \( x_{1} = 1 \)?

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