
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 geometric sequence $\left\{a_{n}\right\}$ with a common ratio $q \gt 1$, satisfying: $a_{2}+a_{3}+a_{4}=28$, and $a_{3}+2$ is the arithmetic mean of $a_{2}$ and $a_{4}$.

(1) Find the general formula for the sequence $\left\{a_{n}\right\}$.

(2) If $b_{n}=a_{n}\log_{\frac{1}{2}}a_{n}$ and $S_{n}=b_{1}+b_{2}+\ldots +b_{n}$, find the smallest positive integer value of $n$ that satisfies $S_{n}+n\cdot 2^{n+1} \gt 62$.

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