
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 $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, sequences $\{a_n\}$ and $\{b_n\}$ are defined as $a_n=\lfloor 10^n x \rfloor - 10\lfloor 10^{n-1} x \rfloor$ and $b_n=\left\lfloor \frac{a_n+1}{k+1} \right\rfloor - \left\lfloor \frac{a_n+1}{k+1.01} \right\rfloor$, where $k\in \mathbb{N}$, $k < 10$. Let $S_n$ be the sum of the first $n$ terms of sequence $\{b_n\}$. When $x= \frac{1}{7}$ and $k=7$, calculate $S_{100}$.

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