
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 \( a_{k}=\frac{2^{k}}{3^{2^{k}}+1} \) for \( k \in \mathbf{N} \). Define \( A = a_{0} + a_{1} + \cdots + a_{9} \) and \( B = a_{0} a_{1} \cdots a_{9} \). Find the value of \( \frac{A}{B} \).

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