
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 that the sequence $\{a_n\}$ is an arithmetic sequence with a common difference $d \neq 0$, and the sequence formed by some terms of $\{a_n\}$, $$a_{b_{1}}, a_{b_{2}}, \ldots, a_{b_{n}}, \ldots$$ is a geometric sequence, where $b_1=1$, $b_2=5$, $b_3=17$.  
(1) Find the general formula for the sequence $\{b_n\}$.  
(2) Let $T_n = C_{n}^{1}b_1 + C_{n}^{2}b_2 + C_{n}^{3}b_3 + \ldots + C_{n}^{n}b_n$, find $T_n$.

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