
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]


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Small and medium-sized enterprises are an important part of the national economy. A small and medium-sized enterprise is preparing to invest special funds in technological innovation to enhance its competitiveness. According to the plan, an investment of $8$ million yuan in special funds in the current year can generate sales revenue of $40$ million yuan. In the following years, the special funds invested each year are half of the previous year, and the sales revenue is $80$ million yuan more than the previous year. At the same time, when the expected investment in special funds is less than $20$ million yuan, an investment of $20$ million yuan is made, and the sales revenue is equal to the sales revenue of the previous year.

$(1)$ Let the special funds invested in the $n$th year (the current year is the first year) be $a_{n}$ million yuan, and the sales revenue be $b_{n}$ million yuan. Please write down the expressions for $a_{n}$ and $b_{n}$.

$(2)$ After how many years at least, can the total sales revenue exceed the total investment in special funds?

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