
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 an arithmetic sequence $\{a_n\}$ and a geometric sequence $\{b_n\}$ with a common ratio of $q$ ($n, q \in \mathbb{N}^*$), let $S_n$ and $T_n$ be the sum of the first $n$ terms of $\{a_n\}$ and $\{b_n\}$, respectively. If $T_{2n}+1=S_{q^n}$, then $a_n= \boxed{\text{2n-1}}$.

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