
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 that for all $n \in \mathbb{N}^{*}$, the function of $x$, $f_{n}(x) = x + (1 - a_{n})\ln x$ where $n < x < n+1$, is not monotonic, where $a_{n} (n = 1, 2, \ldots, k, \ldots)$ is a constant. Define $\lfloor x \rfloor$ as the largest integer not greater than the real number $x$. Let $b_{n} = \left\lfloor \sqrt[3]{a_{n}} \right\rfloor$, and denote the sum of the first $n$ terms of the sequence $ \{b_{n}\}$ as $S_{n}$. Calculate the value of $S_{100}$.

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