
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 the sequence $\{a\_n\}$ is an arithmetic sequence with a common difference $d \neq 0$, and $a\_2$, $a\_6$, $a\_22$ form a geometric sequence, $a\_4 + a\_6 = 26$.

1. Find the general term formula for the sequence $\{a\_n\}$:
2. Let $b\_n = 2^{n-1} \cdot a\_n$, find the sum of the first $n$ terms $T\_n$ for the sequence $\{b\_n\}$.

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