
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
For a finite sequence \( B = (b_1, b_2, \dots, b_n) \), the Cesaro sum is defined as
\[
\frac{S_1 + \cdots + S_n}{n},
\]
where \( S_k = b_1 + \cdots + b_k \) and \( 1 \leq k \leq n \).

If the Cesaro sum of the 99-term sequence \( (b_1, \dots, b_{99}) \) is 800, what is the Cesaro sum of the 100-term sequence \( (10, b_1, \dots, b_{99}) \)?

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