
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 a geometric sequence $\{a_n\}$ with the first term as 1 and the common ratio as $q$, and its sum of the first $n$ terms is $S_n$;  
(1) If $S_3=3$ and $S_6=-21$, find the common ratio $q$;  
(2) If $q>0$, and $T_n=a_1+a_3+\ldots+a_{2n-1}$, find $\lim_{n \to \infty} \frac{S_n}{T_n}$.

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