
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 the sum of the first $n$ terms is denoted as $S_n$. If $a_4 = -12$ and $a_8 = -4$,

(Ⅰ) Find the general formula for the nth term, $a_n$, of the sequence $\{a_n\}$;

(Ⅱ) Determine the minimum value of $S_n$ and the corresponding value of $n$;

(Ⅲ) Form a new sequence $\{b_n\}$ by taking terms $$a_{1}, a_{2}, a_{4}, a_{8}, \ldots, a_{2^{n-1}}, \ldots$$ from the sequence $\{a_n\}$ in sequence, and find the sum of the first $n$ terms of $\{b_n\}$.

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