
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 $\{a_{n}\}$ is a sequence of positive numbers in arithmetic progression, with the sum of the first $n$ terms denoted as $S_{n}$, and $a_{1}=1$. Choose one of the following three conditions as known information to answer the following questions:<br/>$(Ⅰ)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(Ⅱ)$ Let $\{b_{n}\}$ be a sequence such that ${b}_{n}=2{a}_{n}+{2}^{{a}_{n}}$, find the sum of the first $n$ terms of $\{b_{n}\}$.<br/>Condition 1: $a_{5}-a_{3}=2$; Condition 2: $a_{2}a_{3}=6$; Condition 3: $S_{5}=15$.<br/>Note: If multiple conditions are chosen to answer separately, the first answer will be scored.

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