
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
Consider the following number array: $(2)$, $(2^2, 2^3)$, $(2^3, 2^4, 2^5)$, …, $(2^n, 2^{n+1},…, 2^{2n-1})$. Let the sum of all elements in the $n$-th parenthesis be denoted as $a_n$.  
(I) Find a general formula for the sequence $\{a_n\}$.  
(II) Let $b_n = (-1)^n \cdot n^2 + \log_2(4^n - a_n)$, calculate the sum of the first $100$ terms $S_{100}$ of the sequence $\{b_n\}$.

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