
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
Given a periodic sequence $\{x_n\}$ such that $x_n = |x_{n-1} - x_{n-2}|$ for $n \geq 3$, and $x_1 = 1$, $x_2 = a \geq 0$, find the sum of the first 2008 terms of the sequence when the period of the sequence is minimized.


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