
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 sequence $\{a_n\}$, let $S_n$ denote the sum of its first $n$ terms. Define $T_n = \frac{S_1 + S_2 + \dots + S_n}{n}$ as the "ideal number" of the sequence $a_1, a_2, \dots, a_n$. If the "ideal number" of the sequence $a_1, a_2, \dots, a_{502}$ is $2012$, calculate the "ideal number" of the sequence $2, a_1, a_2, \dots, a_{502}$.

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