
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
Four people, A, B, C, and D, jointly registered a company, with each contributing $50$ thousand dollars as start-up capital for production. By the end of the year, the capital had increased by $50\%$. It is expected that the annual growth rate of the capital will be the same as the first year in the future. The four people decided that starting from the first year, the company would distribute $60$ thousand dollars in dividends at the end of each year, and invest all remaining funds in production for the next year. Let the remaining funds after dividends at the end of the $n$th year be denoted as $a_{n}$ thousand dollars.<br/>$(1)$ Find $a_{1}$ and $a_{2}$, and write the relationship between $a_{n+1}$ and $a_{n}$;<br/>$(2)$ After how many years will the remaining funds after dividends in the company be no less than $1200$ thousand dollars?<br/>$(\text{Round the number of years to the nearest integer, reference data: }1.5^{5}\approx 7.59, 1.5^{6}\approx 11.39)$

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