
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 a new sequence $\{b_n\}$, where $b_n = [\log_10 a_n]$, and $[x]$ denotes the largest integer not exceeding $x$ (e.g., $[0.9] = 0$ and $[\log_{10} 99] = 1$).

(I) Find $b_1$, $b_{11}$, and $b_{101}$;

(II) 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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