
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 \( a_{1}, a_{2}, \cdots \) and \( b_{1}, b_{2}, \cdots \) be two arithmetic sequences, with their sums of the first \( n \) terms denoted as \( A_{n} \) and \( B_{n} \) respectively. It is known that for all \( n \in \mathbf{N} \),
$$
\frac{A_{n}}{B_{n}} = \frac{2n-1}{3n+1}.
$$

Determine the expression for \( \frac{a_{n}}{b_{n}} \) for all \( n \in \mathbf{N} \).

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