
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 a sequence $\{a_n\}$ where each term is a positive number and satisfies the relationship $a_{n+1}^2 = ta_n^2 +(t-1)a_na_{n+1}$, where $n\in \mathbb{N}^*$. 

(1) If $a_2 - a_1 = 8$, $a_3 = a$, and the sequence $\{a_n\}$ is unique:
   ① Find the value of $a$.
   ② Let another sequence $\{b_n\}$ satisfy $b_n = \frac{na_n}{4(2n+1)2^n}$. Is there a positive integer $m, n$ ($1 < m < n$) such that $b_1, b_m, b_n$ form a geometric sequence? If it exists, find all possible values of $m$ and $n$; if it does not exist, explain why.

(2) If $a_{2k} + a_{2k-1} + \ldots + a_{k+1} - (a_k + a_{k-1} + \ldots + a_1) = 8$, with $k \in \mathbb{N}^*$, determine 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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