
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 sequence $\left\{ a_{n} \right\}$ satisfying $a_{1} > 0$ and $a_{n} = \frac{a_{1} + a_{n-1}}{1 - a_{1} a_{n-1}}$ for $n \geq 2$, and having the smallest positive period of 2008, determine the value of $a_{1}$.

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