
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
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$, where $n\in\mathbb{N}^*$ and the common difference $d\neq 0$. Given that $S_3=15$ and that $a_1$, $a_4$, $a_{13}$ form a geometric sequence.

(Ⅰ) Find the general formula for the sequence $\{a_n\}$;  
(Ⅱ) Let $b_n=a_{2^n}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$.

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