
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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Let the sum of the first $n$ terms of the arithmetic sequence $\{a_n\}$ be $\{S_n\}$, and it is given that $S_4=4S_2$, $a_{2n}=2a_n+1$.

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

(Ⅱ) Let the sum of the first $n$ terms of the sequence $\{b_n\}$ be $T_n$, and it is given that $T_n+ \frac{a_n+1}{2^n}=\lambda$ (where $\lambda$ is a constant). Let $c_n=b_{2n}$ ($n\in\mathbb{N}^*$), find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted as $R_n$.

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