
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 arithmetic sequence $\{a_{n}\}$ satisfies $a_{1}=1$ and $a_{2}+a_{6}=14$. The geometric sequence $\{b_{n}\}$ with positive terms satisfies $b_{1}=2$ and $b_{3}=8$. Find:
1. The general term formulas $a_{n}$ and $b_{n}$ for both sequences.
2. The sum $T_{n}$ of the first $n$ terms of the sequence $\{(a_{n}+1)\cdot b_{n}\}$.

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