
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 ${{a_n}}(n = 1,2,...,n)$, let ${S_n}$ be its partial sum. The "Cesàro sum" is defined as $\frac{{{S_1} + {S_2} + \cdots {S_n}}}{n}$. Now, the "Cesàro sum" of a 500-term sequence $({a_1}),({a_2}),...,({a_{500}})$ is 2004. Determine the "Cesàro sum" of the 501-term sequence $(2),({a_1}),({a_2}),...,({a_{500}})$.

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