
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
The sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ is $S_n$, and $a_1=1$, $S_7=28$. Let $b_n=\lfloor \lg a_n \rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, such as $\lfloor 0.9 \rfloor=0$, $\lfloor \lg 99 \rfloor=1$. Then, the sum of the first 1000 terms of the sequence $\{b_n\}$ is \_\_\_\_\_\_.

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