
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
Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic sequence ${a_{n}}$, with $a_{1}=1$ and $S_{7}=28$. Define $b_{n}=\lfloor \log_{10} a_{n} \rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer not exceeding $x$ (e.g., $\lfloor 0.9 \rfloor=0$ and $\lfloor \log_{10} 99 \rfloor=1$).

(1) Find $b_{1}$, $b_{11}$, and $b_{101}$;

(2) Find the sum of the first $1000$ terms of the sequence ${b_{n}}$.

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