
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 $\{a_n\}$ with the sum of the first $n$ terms denoted as $S_n$, the common difference $d \neq 0$, and $S_3 + S_5 = 50$, it is also known that $a_1$, $a_4$, and $a_{13}$ form a geometric sequence.  
(1) Find the general formula for the sequence $\{a_n\}$.  
(2) If elements are sequentially taken from the sequence $\{a_n\}$ as the 2nd, 4th, 8th, ..., $2^n$th, ..., and arranged in the original order to form a new sequence $\{b_n\}$, with the sum of the first $n$ terms of this sequence denoted as $T_n$, find the expression for $T_n$.

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