
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 a common difference of $2$, and $a_{1}$, $a_{1}+a_{2}$, $2(a_{1}+a_{4})$ forms a geometric sequence.

1. Find the general term formula for the sequence $\{a_{n}\}$.
2. Let $b_{n}=a_{n}+2^{n-1}$, find the sum of the first $n$ terms, $S_{n}$, for the sequence $\{b_{n}\}$.

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