
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 that $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$, with $a_1=3$, and that $2S_n=a_{n+1}-3$ for all positive integers $n$.  
(1) Find the general formula for the $n$-th term of the sequence $\{a_n\}$.  
(2) For positive integers $i$, $j$, and $k$ with $i < j < k$, it is known that $λa_j$, $6a_i$, and $μa_k$ form an arithmetic sequence. Determine the values of the positive integers $λ$ and $μ$.  
(3) Let $\{b_n\}$ be a sequence whose sum of the first $n$ terms is denoted by $T_n$, and for all positive integers $n$, the following equation holds: $a_1b_n + a_2b_{n-1} + a_3b_{n-2} + \ldots + a_nb_1 = 3^{n+1} - 3n - 3$. Determine all positive integers $n$ that satisfy the equation $\frac{T_n}{a_n} = \frac{1}{3}$.

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