
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 \(P = \left(p_{1}, p_{2}, \cdots, p_{n}\right)\), the Caesar sum (named after a mathematician Caesar) is defined as \(\frac{s_{1}+s_{2}+\cdots+s_{n}}{n}\), where \(s_{k} = p_{1} + p_{2} + \cdots + p_{k}\) for \(1 \leq k \leq n\). If a sequence of 99 terms \(\left(p_{1}, p_{2}, \cdots, p_{99}\right)\) has a Caesar sum of 1000, determine the Caesar sum of the 100-term sequence \(\left(1, p_{1}, p_{2}, \cdots, p_{99}\right)\).

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