
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 $\{a_n\}$ where $a_1 = 1$ and $a_2 + a_6 = 14$:
1. Find the general formula for the sequence $\{a_n\}$.
2. Let another sequence $\{b_n\}$ satisfy the following condition:
   $$\frac{b_1}{2} + \frac{b_2}{2^2} + \frac{b_3}{2^3} + \ldots + \frac{b_n}{2^n} = a_n + n^2 + 1$$
   Find the sum of the first n terms of sequence $\{b_n\}$, denoted as $T_n$.

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