
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 an arithmetic sequence $\{a_{n}\}$ with a non-zero common difference, where $a_{2}$ is the geometric mean of $a_{1}$ and $a_{4}$, and $a_{5}+a_{6}=11$.<br/>$(Ⅰ)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(Ⅱ)$ Choose one of the following conditions as known to find the sum of the first $n$ terms of the sequence $\{b_{n}\}$.<br/>①${b_n}={a_n}⋅{2^{{a_n}}}$;<br/>②${b_n}=\frac{{{a_{2n}}^2}}{{({2{a_n}-1})({2{a_n}+1})}}$.<br/>Note: If multiple conditions are chosen and answered separately, the first answer will be scored.

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