
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

The sequence \( \{a_n\} \) satisfies \( a_0 = 0, a_1 = 1, a_{n+2} = a_{n+1} + a_n \) (for \( n \in \mathbb{N} \)). 

The sequence \( \{b_n\} \) satisfies \( b_1 \in \mathbb{R} \), \( b_n = \frac{a_{n-2} b_1 - a_{n-1}}{a_n - a_{n-1} b_1} \) (for \( n = 2, 3, 4, \ldots \)).

If \( b_{2009} = \frac{1}{b_1} + 1 \), then \( b_1 = \qquad \).

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