
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
Mianning Lingshan Temple is a national 4A-level tourist attraction and one of the most popular tourist attractions in Liangshan Prefecture. Lingshan Temple is renowned as "the first spirit in the world," "the first mountain in southern Sichuan," and "the first temple in western Sichuan," and it is always bustling with incense all year round. The number of tourists visiting Lingshan Temple has been increasing rapidly year by year. According to statistics, the number of tourists visiting Lingshan Temple from 2021 to 2023 is shown in the table below:

| Year   | 2020 | 2021 | 2022 |
|--------|------|------|------|
| Year Code $x$ | 1 | 2 | 3 |
| Number of Tourists $y$ (in 10,000 people) | 12 | 18 | 27 |

Based on the above data, there are two function models to choose from for the annual number of tourists $y$ at Lingshan Temple and the year code $x$ (Note: the year code for 2020 is $x=1$, for 2021 is $x=2$, and so on): 
1. $y=ka^{x}$ ($k>0$, $a>1$)
2. $y=m\sqrt{x}+n$ ($m>0$)

$(1)$ Determine which function model is more appropriate (no calculation required, just give a brief explanation), and find the analytical expression of the chosen function model.
$(2)$ Approximately in which year will the annual number of tourists at Lingshan Temple be about 3 times the number of tourists in 2021? (Reference data: $\sqrt{2}\approx1.41$, $\sqrt{3}\approx1.73$, $\log{2}\approx0.30$, $\log{3}\approx0.48$)

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