
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 $\{a_n\}$ with the sum of its first $n$ terms represented by $S_n$, such that $a_5 + a_6 = 24$ and $S_{11} = 143$. For another sequence $\{b_n\}$ with the sum of its first $n$ terms denoted by $T_n$, the following relationship holds: $$2^{a_n - 1} = \lambda T_n - (a_1 - 1) \quad (n \in \mathbb{N}^*)$$
(Ⅰ) Find the general term formula for the sequence $\{a_n\}$ and the sum of the first $n$ terms for the sequence $\left\{\frac{1}{a_n a_{n+1}}\right\}$;
(Ⅱ) Does there exist a non-zero real number $\lambda$ such that the sequence $\{b_n\}$ forms a geometric sequence? Justify your reasoning.

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