
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 monotonically increasing geometric sequence \(\left\{a_{n}\right\}\) that satisfies \(a_{2}+a_{3}+a_{4}=28\), and \(a_{3}+2\) is the arithmetic mean of \(a_{2}\) and \(a_{4}\):

1. Find the general term formula for the sequence \(\left\{a_{n}\right\}\).
2. Let \(b_{n} = a_{n} \log_{\frac{1}{2}} a_{n}\) and \(S_{n} = b_{1} + b_{2} + \cdots + b_{n}\). Find the smallest positive integer \(n\) such that \(S_{n} + n \cdot 2^{n+1} > 50\).

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