
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]


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Given that the sequence $\{a_n\}$ forms a geometric sequence, and $a_n > 0$.

(1) If $a_2 - a_1 = 8$, $a_3 = m$. ① When $m = 48$, find the general formula for the sequence $\{a_n\}$. ② If the sequence $\{a_n\}$ is unique, find the value of $m$.

(2) If $a_{2k} + a_{2k-1} + \ldots + a_{k+1} - (a_k + a_{k-1} + \ldots + a_1) = 8$, where $k \in \mathbb{N}^*$, find the minimum value of $a_{2k+1} + a_{2k+2} + \ldots + a_{3k}$.

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