
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
Consider a geometric sequence with the first term equal to 2 and the common ratio denoted by $q$ ($q>0$). Let $S_n$ represent the sum of the first $n$ terms of the sequence, and let $T_n=a_2+a_4+a_6+\ldots+a_{2n}$.

(1) Find $S_n$.
(2) Calculate $$\lim_{n \to \infty} \frac{S_n}{T_n}$$.

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