
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 $\{a_n\}$ whose terms are all positive numbers, and the sum of the first $n$ terms is $S_n$, it satisfies $(p-1)S_n=p^9-a_n$, where $p$ is a positive constant and $p\neq 1$.

(Ⅰ) Find the general formula for the sequence $\{a_n\}$;

(Ⅱ) Let $b_n= \frac{1}{9-\log_{p}a_n}$ $(n\in\mathbb{N}^*)$, find the sum of the first $n$ terms $T_n$ of the sequence $\{b_nb_{n+1}\}$;

(Ⅲ) Let $c_n=\log_{2}a_{2n-1}$, the sum of the first $n$ terms of the sequence $\{c_n\}$ is $H_n$. If $H_n$ has a maximum value when $n\in\mathbb{N}^*$, find the range of $p$ and the maximum value.

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