
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 a sequence $\{a_n\}$ where $a_1=3$, $a_2=5$, and the sum of the first $n$ terms $S_n$ satisfies $S_n+S_{n-2}=2S_{n-1}+2^{n-1}$ for $n\geqslant 3$.

(Ⅰ) Find the general formula for the sequence $\{a_n\}$;  
(Ⅱ) Let $b_n=\log_2\left(\frac{256}{a_{2n}-1}\right)$ for $n\in\mathbb{N}^*$, and let the sum of the first $n$ terms of the sequence $\{b_n\}$ be $S_n$. For what value of $n$ does $S_n$ reach its maximum value, and what is the maximum value?

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