
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 sequence of positive integers \( \{a_n\} \) with its first \( n \) terms' sum denoted by \( \{S_n\} \). Let the first \( n \) terms' sum of the sequence \( \{S_n + 1\} \) be denoted by \( \{T_n\} \). Is there a unique sequence \( \{a_n\} \) such that the sequence \( \{T_n\} \) is the same as the sequence \( \{S_{n+1} - 1\} \)? If it exists, find the general term of the sequence \( \{a_n\} \). If it does not exist, find the value of \( \lim_{n \to \infty} \frac{a_2 + a_4 + \cdots + a_{2n}}{a_1 + a_3 + \cdots + a_{2n-1}} \).

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