
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 the arithmetic sequence $\{a\_n\}$ and the geometric sequence $\{b\_n\}$ with a common ratio of $q(n, q∈N^*)$, let $S\_n$ and $T\_n$ denote the sum of the first $n$ terms of $\{a\_n\}$ and $\{b\_n\}$, respectively. If $T_{2n} + 1 = S_{q^n}$, find the expression for $a\_n$.

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