
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 $\{a_n\}$ be a sequence of positive numbers with $b_n$ representing the sum of the first $n$ terms of $\{a_n\}$. The product of the first $n$ terms of the sequence $\{b_n\}$ is denoted by $c_n$, and $b_n + c_n = 1$. What is the number in the sequence $\left\{\frac{1}{a_n}\right\}$ that is closest to 2009?

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