
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 finite sequence $(a\_1, a\_2, a\_3, ..., a\_k) (k \in \mathbb{N}^*, k \geq 3)$ where all elements are positive numbers and satisfy the following conditions:
$(1) a\_1 = a\_k$;
$(2) a\_n + \frac{2}{a\_n} = 2a_{n+1} + \frac{1}{a_{n+1}} (n = 1, 2, 3, ..., k - 1)$.

(I) If $k = 3$ and $a\_1 = 2$, find the sequence.
(II) If $k = 4$, find the set of all possible values of $a\_1$.
(III) If $k$ is even, find the maximum value of $a\_1$ (represented by $k$).

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