
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 that $\\{a_{n}\\}$ is a geometric sequence with $a_{1}=2$, and that $a_{2}$, $a_{3}+2$, $a_{4}$ form an arithmetic sequence.
(Ⅰ) Find the general formula for $\\{a_{n}\\}$.
(Ⅱ) Let $b_{n}=2na_{n}$, and the sum of the first $n$ terms of $\\{b_{n}\\}$ be $S_{n}$. Define $g(n)= \frac {2n^{2}-9n+7}{S_{n}-4}$ ($n\geqslant 2,n∈\mathbb{N}^{\*}$), find the value of a positive integer $k$ such that $g(k)\geqslant g(n)$ for all $n\geqslant 2$.

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