
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]


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For a finite sequence $B = (b_1, b_2, \dots, b_m)$ of numbers, the Cesaro sum of $B$ is defined to be
\[\frac{T_1 + T_2 + \dots + T_m}{m},\]
where $T_k = b_1 + b_2 + \dots + b_k$ for $1 \leq k \leq m$.

If the Cesaro sum of the 50-term sequence $(b_1, b_2, \dots, b_{50})$ is 500, what is the Cesaro sum of the 51-term sequence $(2, b_1, b_2, \dots, b_{50})$?

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