
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
Given a finite sequence $D$: $a\_1$, $a\_2$, ..., $a\_n$, where $S\_n$ represents the sum of the first $n$ terms of the sequence $D$, define $\frac{S\_1 + S\_2 + ... + S\_n}{n}$ as the "De-Guang sum" of $D$. If a 99-term sequence $a\_1$, $a\_2$, ..., $a\_99$ has a "De-Guang sum" of $1000$, find the "De-Guang sum" of the 100-term sequence $8$, $a\_1$, $a\_2$, ..., $a\_99$.

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