
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
Let the sequence $\{a_n\}$ have a partial sum $S_n$ for its first $n$ terms. Define $T_n = \frac{S_1 + S_2 + \ldots + S_n}{n}$ as the "ideal number" for the sequence $a_1, a_2, \ldots, a_n$. Given that the "ideal number" for the sequence $a_1, a_2, \ldots, a_{500}$ is 2004, determine the "ideal number" for the sequence 12, $a_1, a_2, \ldots, a_{500}$.

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