
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 $\{a\_n\}$ where all terms are positive numbers, and it satisfies the equation $a_{n+1}^{2}-a_{n+1}a_{n}-2a_{n}^{2}=0 (n\in N^{*})$, also $a_{3}+2$ is the middle term of the arithmetic sequence with terms $a_{2}$ and $a_{4}$.
(1) Find the general term formula $a_{n}$ of the sequence;
(2) If $b_{n}=a_{n}\log _{\frac{1}{2}}a_{n}$, $S_{n}=b_{1}+b_{2}+…+b_{n}$, find the smallest positive integer value of $n$ such that $S_{n}+n⋅2^{n+1} > 50$.

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