
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
Lao Li is a well-known expert in fish farming in the local area. He is preparing to contract a fish farm and sign a contract. After calculation and research, it is predicted that the growth rate of fish weight in the first year is $200\%$, and the weight growth rate in the following years is half of the previous year's growth rate. However, due to the growth of fish, the oxygen content in the water will decrease, causing the fish to grow slowly. To ensure the normal growth of fish, the oxygen content in the water must be maintained above a certain level. It is known that the oxygen content in the water is 8 units in the first year. According to the information obtained from the scientific and technological personnel, it is understood that for fish to grow normally, the oxygen content in the water is 4.5 units in the third year. The function model of oxygen content $y$ with respect to year $x$ is $y=ka^{x}(k \gt 0,0 \lt a \lt 1)$. When the oxygen content is less than $\frac{81}{32}$ units, although the fish will continue to grow, 5% of the total weight will be lost. When the total weight in a certain year starts to decrease compared to the previous year, it is time to fish in a timely manner, which is also the most suitable time to sign the contract.<br/>$(1)$ Find the relationship model function of the oxygen content;<br/>$(2)$ Determine in which year the fish growth will slow down and losses will occur due to the relationship with the oxygen content;<br/>$(3)$ Find the relationship formula between the total weight of fish in the $(n+1)$th year $a_{n+1}$ and the $n$th year $a_{n}$ (without proving the relationship formula, $n$ is an integer), and determine how many years is the most suitable time to sign the contract.

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