
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 positive integer \( k \), a sequence \( \{a_n\} \) satisfies the following conditions:
\[ a_1 = 3, \quad a_{n+1} = \left(3^{\frac{2}{2x-1}} - 1\right) S_n + 3 \quad (n \in \mathbb{Z}_+) \]
where \( S_n \) is the sum of the first \( n \) terms of the sequence \( \{a_n\} \).

Let 
\[ b_n = \frac{1}{n} \log_3 (a_1 a_2 \ldots a_n) \quad (n \in \mathbb{Z}_+) \]
and define
\[ T_k = \sum_{i=1}^{2k} \left| b_i - \frac{3}{2} \right|. \]

If \( T_k \in \mathbb{Z}_+ \), find all possible values of \( k \).

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