
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 an arithmetic sequence $\{a_n\}$ with the first term $a_1=10$ and common difference $d=2$, and a geometric sequence $\{b_n\}$ with the first term $b_1=1$ and common ratio $r=2$, where $n \in \mathbb{N}^*$.  
(1) Find the general formula for the $n$-th term of sequences $\{a_n\}$ and $\{b_n\}$.  
(2) Let the side length of the $n$-th square be $C_n = \min\{a_n, b_n\}$. Calculate the sum of areas $S_n$ of the first $n$ squares. (Note: $\min\{a, b\}$ denotes the minimum value between $a$ and $b$.)

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