
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 the following conditions:①$\frac{1}{{{S_1}}}+\frac{1}{{{S_2}}}+\frac{1}{{{S_3}}}+⋯+\frac{1}{{{S_n}}}=\frac{n}{{n+1}}$; ②$a_{1}=2$,$na_{n+1}=2S_{n}$; ③$4S_{n}=a_{n}^{2}+2a_{n}$, choose one of these three conditions to fill in the blank line below and answer the following questions.<br/>For a positive sequence $\{a_{n}\}$ with the sum of its first $n$ terms being $S_{n}$, _______,$n\in N^{*}$.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ If the sequence $\{b_{n}\}$ satisfies ${b_n}={2^{\frac{{{a_n}}}{2}}}$, and $\Omega \left(x\right)$ represents the remainder of $x$ divided by $3$, find $Ω({\sum_{i=1}^{2n+1}{{b_i}}})$.<br/>Note: If multiple conditions are chosen for separate solutions, the first solution will be scored.

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