
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
If sequence $\{x_n\}$ satisfies $x_{n+1} = |x_n - x_{n-1}|$ (for $n \geq 2$, $n \in \mathbb{N}$) and is given with $x_1 = 1$, $x_2 = a$ (where $a \in \mathbb{R}, a \neq 0$), determine the sum of the first 2010 terms of the sequence $\{x_n\}$ when the period of $\{x_n\}$ is the smallest possible.

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