
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 geometric sequence $\{c_n\}$ satisfying $c_{n+1} + c_n = 10 \cdot 4^{n-1}$ $(n \in \mathbb{N}^*)$, let $S_n$ be the sum of the first $n$ terms of a sequence $\{a_n\}$, where $a_n = \log_2 c_n$,
(Ⅰ) Find $a_n$ and $S_n$;
(Ⅱ) Given another sequence $\{b_n\}$ defined by $b_n = \frac{1}{4S_n - 1}$ and $T_n$ as the sum of the first $n$ terms of $\{b_n\}$, is there a pair of positive integers $m$, $k$ ($1 < m < k$) such that $T_1$, $T_m$, $T_k$ form a geometric sequence? If there is, find all possible values of $m$ and $k$; if not, explain the reason.

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