
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 that the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, and $S_{n+1}=S_{n}+a_{n}+1$, _____. Please choose one of the following conditions to supplement the above question: ① $a_{3}+a_{15}=20$; ② $a_{2}$, $a_{5}$, $a_{11}$ form a geometric sequence; ③ $S_{20}=230$, and answer the following questions:<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ If $b_{n}=a_{n}-1$, find the sum of the first $n$ terms of the sequence $\{2^{n}\cdot b_{n}\}$.<br/>Note: If multiple conditions are chosen to answer separately, scoring will be based on the first answer.

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