
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 $\{a_{n}\}$ be a sequence with the sum of its first $n$ terms denoted as $S_{n}$, $a_{1}=1$, and $a_{n+1}=\left\{\begin{array}{l}{2{a}_{n}, \text{if } n \text{ is odd}}\\{{a}_{n}+1, \text{if } n \text{ is even}}\end{array}\right.$.<br/>$(Ⅰ)$ Find the values of $a_{2}$ and $a_{3}$, and determine the general formula for the sequence $\{a_{n}\}$;<br/>$(Ⅱ)$ Determine if there exists a positive integer $n$ such that $\frac{S_{n}}{a_{n}}\in \mathbb{Z}$. If such $n$ exists, find all the values of $n$ that satisfy the condition; if not, explain the reason.

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