
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 of $B$ is defined to be
\[\frac{T_1+\cdots+T_n}{n},\]where $T_k=b_1+\cdots+b_k$ for $1\leq k\leq n$.

If the Cesaro sum of the 49-term sequence $(b_1,\dots,b_{49})$ is 500,  what is the Cesaro sum of the 51-term sequence $(b_1,\dots,b_{49}, 0, 0)$?

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