
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
Consider a sequence ${a_n}$ with its first $n$ terms sum denoted as $S_n$. The sequence is defined by $a_1=1$ and $a_{n+1}=2S_n+1$ for $n\geqslant 1$.

1. Find the general term formula for ${a_n}$.
2. Let ${b_n}$ be an arithmetic sequence with positive terms and its first $n$ terms sum denoted as $T_n$. Suppose $T_3=15$, and that the sequences ${a_1+b_1}$, ${a_2+b_2}$, ${a_3+b_3}$ form a geometric sequence. Determine $T_n$.

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