
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
At the end of $1997$, the desert area in a certain region was $9\times 10^{5}hm^{2}$ (note: $hm^{2}$ is the unit of area, representing hectares). Geologists started continuous observations from $1998$ to understand the changes in the desert area of this region. The observation results at the end of each year are recorded in the table below:

| Year | Increase in desert area compared to the original area (end of year) |
|------|--------------------------------------------------------------------|
| 1998 | 2000                                                               |
| 1999 | 4000                                                               |
| 2000 | 6001                                                               |
| 2001 | 7999                                                               |
| 2002 | 10001                                                              |

Based on the information provided in the table, estimate the following:
$(1)$ If no measures are taken, approximately how much will the desert area of this region become by the end of $2020$ in $hm^{2}$?
$(2)$ If measures such as afforestation are taken starting from the beginning of $2003$, with an area of $8000hm^{2}$ of desert being transformed each year, but the desert area continues to increase at the original rate, in which year-end will the desert area of this region be less than $8\times 10^{5}hm^{2}$ for the first time?

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