
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 ${\{a_n\}}$ with the sum of its first $n$ terms denoted as $S_n$, let $T_n = \frac{S_1 + S_2 + \cdots + S_n}{n}$. We define $T_n$ as the "ideal number" of the terms $a_1, a_2, \ldots, a_n$. It is known that the "ideal number" of the terms $a_1, a_2, \ldots, a_{20}$ is $21$. Find the "ideal number" of the terms $3, a_1, a_2, \ldots, a_{20}$.

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