
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
The sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, $a_{1}=1$, and the point $(S_{n}, a_{n+1})$ lies on the line $y=2x+1$. <br/>$(1)$ Find the general formula for $\{a_{n}\}$; <br/>$(2)$ For an arithmetic sequence $\{b_{n}\}$ with positive terms, the sum of the first $n$ terms is $T_{n}$, and $T_{3}=15$. Also, $a_{1}+b_{1}$, $a_{2}+b_{2}$, $a_{3}+b_{3}$ form a geometric sequence. Find $T_{n}$.

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