
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 \( S_{1}=2^{2}, S_{2}=2^{4}, \cdots, S_{n}=2^{2n} \). Construct a rectangle \( OABC \) as follows: On the \( x \)-axis, mark points \( O, A_{1}, A_{2}, \cdots, A_{n} \); on the \( y \)-axis, mark points \( C_{1}, C_{2}, \cdots, C_{n} \); such that for any positive integer \( k (1 \leq k \leq n) \), we have:
\[ A_{k-1}A_{k} = x_{k}, \; C_{k-1}C_{k} = y_{k}, \]
where \( A_{0}=C_{0}=O, A_{n}=A, C_{n}=C \), and \( x_{k}, y_{k} \in \mathbf{Z}_{+} \), and \( x_{k}y_{k} = S_{k} \). Find:
(1) The maximum area of the rectangle \( OABC \) and the corresponding \( \left(x_{1}, x_{2}, \cdots, x_{n}\right) \) and \( \left(y_{1}, y_{2}, \cdots, y_{n}\right) \);
(2) The minimum area of the rectangle \( OABC \) and the corresponding \( \left(x_{1}, x_{2}, \cdots, x_{n}\right) \) and \( \left(y_{1}, y_{2}, \cdots, y_{n}\right) \).

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