
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 the geometric sequence $\{a\_n\}$ satisfying $a\_1=2$, $a\_2=4(a\_3-a\_4)$, and the sequence $\{b\_n\}$ satisfying $b\_n=3-2\log\_2 a\_n$.
(I) Find the general terms of the sequences $\{a\_n\}$ and $\{b\_n\}$;
(II) Let $c\_n=\frac{a\_n}{b\_n}$, find the sum of the first $n$ terms of the sequence $\{c\_n\}$, denoted as $T\_n$;
(III) If $\lambda > 0$, find the range of $k$ such that $2\lambda^2-k\lambda+2 > a\_{2n}b\_n$ holds for all positive integers $n$.

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