
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 a positive sequence $\left\{a_n\right\}$ with the sum of the first $n$ terms denoted as $S_n$, and $a_1=2$, $a_n^2=4S_{n-1}+4n$ $(n\geqslant 2)$ ...

(Ⅰ) Find the general formula for the sequence $\left\{a_n\right\}$;

(Ⅱ) Calculate the value of $a_2+a_5+a_8+\ldots+a_{89}$.

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