
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 \(a_1, a_2, a_3, \ldots\) be a sequence of real numbers satisfying
\[a_n = a_{n-1} a_{n+1}\] for all \(n \geq 2\). If \(a_1 = 1 + \sqrt{3}\) and \(a_{1776} = 4 + \sqrt{3}\), determine \(a_{2009}\).

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