
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 ${a\_n}$, with $a\_1 = 1$ and $S\_7 = 28$. Define $b\_n = [\log_{10} a\_n]$, where $[x]$ denotes the largest integer not greater than $x$ (e.g., $[0.9] = 0$, $[\log_{10} 99] = 1$).

(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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