
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
A reward fund distributes its awards in the following manner: once a year, the total amount of prize money is divided equally into 6 parts, rewarding individuals who have made the most beneficial contributions to humanity in 6 different areas. The total amount of prize money distributed each year is half of the interest earned by the fund in that year, with the other half of the interest being added to the total fund to ensure that the amount of prize money increases annually. Assuming the average annual interest rate of the fund is $r=6.24\%$, and after the awards were distributed in 2000, the total amount of the fund was approximately 21000 million yuan. Let $a_n$ represent the total amount of the fund after the awards have been distributed in the $n$-th year ($n \in \mathbb{N}^*$, with 2000 being the first year).

$(1)$ Express $a_2$ and $a_3$ in terms of $a_1$, and based on the results obtained, deduce the expression for $a_n$;  
$(2)$ Based on the expression for $a_n$, determine whether the prize money for each category exceeded 15 million yuan in 2011, and calculate the total amount of prize money distributed from 2001 to 2011.  
(Reference data: $1.0624^{10}=1.83$, $1.032^{9}=1.32$, $1.0312^{10}=1.36$, $1.032^{11}=1.40$)

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