
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
Let an arithmetic sequence $\{a_n\}$ have the sum of the first $n$ terms denoted by $S_n$, with $S_4 = 4S_2$, and $a_{2n} = 2a_n + 1$.

(1) Determine the general term formula for the sequence $\{a_n\}$;

(2) Let another sequence $\{b_n\}$ have the sum of the first $n$ terms denoted by $T_n$, and $T_n + \frac{a_n+1}{2^n} = \lambda$ (where $\lambda$ is a constant). Define $c_n = b_{2n}$ for $n \in \mathbb{N}^{*}$. 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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