
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 $b_1, b_2, b_3, \dots$ be a sequence of real numbers satisfying the recursion
\[b_n = b_{n-1}b_{n+1}\]
for all $n \ge 2$. If $b_1 = 2 + \sqrt{8}$ and $b_{1776} = 14 + \sqrt{8}$, determine $b_{2009}$.

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