
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 the first $n$ terms denoted as $S\_n$. It is known that $S\_4 = 4S\_2$ and $a\_2n = 2a\_n + 1$.

(1) Find the general term formula for the sequence $\{a\_n\}$.

(2) Given another sequence $\{b\_n\}$ with the sum of the first $n$ terms denoted as $T\_n$. If $T\_n + \frac{a\_n + 1}{2^n} = λ$ (where $λ$ is a constant), let $c\_n = b\_{2n}$ for all positive integers $n$. 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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