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

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