
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 that the sum of the first $n$ terms of the sequence $\{a\_n\}$ is $S\_n$, let $T\_n = \frac{S\_1 + S\_2 + \cdots + S\_n}{n}$. It is known that the "ideal number" of the sequence $a\_1, a\_2, \ldots, a\_502$ is $2012$, determine the "ideal number" of the sequence $2, a\_1, a\_2, \ldots, a\_502$.

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