
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 function $f(x) = \frac{4^x}{4^x + 2}$.
- (I) Find the value of $f(x) + f(1-x)$, where $x \in \mathbb{R}$.
- (II) If the sequence $\{a_n\}$ satisfies $a_n = f(0) + f\left(\frac{1}{n}\right) + f\left(\frac{2}{n}\right) + \ldots + f\left(\frac{n-1}{n}\right) + f(1)$ ($n \in \mathbb{N}^*$), find the general formula for the sequence $\{a_n\}$.
- (III) If the sequence $\{b_n\}$ satisfies $b_n = 2^{n+1} \cdot a_n$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, $S_n$.

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