
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 the arithmetic sequence $\{a_n\}$, and $a_1=1$, $S_7=28$. Define $b_n=\left\lfloor \lg a_n \right\rfloor$, where $\left\lfloor x \right\rfloor$ denotes the greatest integer less than or equal to $x$, for example, $\left\lfloor 0.9 \right\rfloor=0$, $\left\lfloor \lg 99 \right\rfloor=1$.
- (I) Find $b_1$, $b_{11}$, $b_{101}$.
- (II) Calculate 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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