
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 sequence $\{a\_n\}$ with its sum of the first $n$ terms denoted as $S\_n$. For any $n ∈ ℕ^∗$, it is known that $S\_n = 2(a\_n - 1)$.

1. Find the general term formula for the sequence $\{a\_n\}$.
2. Insert $k$ numbers between $a\_k$ and $a_{k+1}$ to form an arithmetic sequence with a common difference $d$ such that $3 < d < 4$. Find the value of $k$ and the sum $T$ of all terms in this arithmetic sequence.

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