
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
We know that every natural number has factors. For a natural number $a$, we call the positive factors less than $a$ the proper factors of $a$. For example, the positive factors of $10$ are $1$, $2$, $5$, $10$, where $1$, $2$, and $5$ are the proper factors of $10$. The quotient obtained by dividing the sum of all proper factors of a natural number $a$ by $a$ is called the "perfect index" of $a$. For example, the perfect index of $10$ is $\left(1+2+5\right)\div 10=\frac{4}{5}$. The closer the "perfect index" of a natural number is to $1$, the more "perfect" we say the number is. If the "perfect index" of $21$ is _______, then among the natural numbers greater than $20$ and less than $30$, the most "perfect" number is _______.

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