
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 sequence $\{a_n\}$ with the first term being $1$, and $S_n$ being the sum of the first $n$ terms of the sequence $\{a_n\}$, it satisfies $S_{n+1}=qS_{n}+1$, where $q > 0$, $n\in\mathbb{N}^*$, and $2a_{2}$, $a_{3}$, $a_{2}+2$ form an arithmetic sequence.  
(Ⅰ) Find the general formula for the sequence $\{a_n\}$.  
(Ⅱ) Let $b_{n}=2a_{n}-\lambda(\log_{2}a_{n+1})^{2}$. If the sequence $\{b_{n}\}$ is increasing, find the range of values for $\lambda$.

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