
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 that \(\{a_{n}\}\) and \(\{b_{n}\}\) are arithmetic sequences (i.e. \(a_{2} - a_{1} = a_{3} - a_{2} = a_{4} - a_{3} = \cdots\) and \(b_{2} - b_{1} = b_{3} - b_{2} = b_{4} - b_{3} = \cdots\)) such that \((3n + 1)a_{n} = (2n - 1)b_{n}\) for all positive integers \(n\). If \(S_{n} = a_{1} + a_{2} + \cdots + a_{n}\) and \(T_{n} = b_{1} + b_{2} + \cdots + b_{n}\), find \(\frac{S_{9}}{T_{6}}\).

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