
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 the binomial expansion $(3x - 1)^{2n} = a_{2n} x^{2n} + a_{2n-1} x^{2n-1} + \cdots + a_{2} x^{2} + a_{1} x + a_{0}$, let $T_{n} = a_{0} + a_{2} + \cdots + a_{2n}$ and $R_{n} = a_{1} + a_{3} + \cdots + a_{2n-1}$, then find $\lim _{n \rightarrow+\infty} \frac{T_{n}}{R_{n}}$.

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