
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
In a certain region, the residential water price in the previous year was $2.8$ yuan/$m^{3}$, with an annual water usage of $am^{3}$. This year, the plan is to reduce the water price to between $2.3$ yuan/$m^{3}$ and $2.6$ yuan/$m^{3}$, while users expect the water price to be $2$ yuan/$m^{3}$. It is calculated that the additional water usage after the price reduction and the difference between the actual water price and the user's expected water price are inversely proportional (with a proportionality coefficient of $k$). It is known that the cost price of water in this region is $1.8$ yuan/$m^{3}$.

$(1)$ Write the analytical expression of the function $y$ of the water department's revenue after the water price reduction this year with respect to the actual water price $x$ (unit: yuan/$m^{3}$). (Revenue $=$ actual water quantity $\times$ (actual water price $-$ cost price))

$(2)$ Suppose $k=0.4a$, what is the minimum water price that can still ensure an increase in the water department's revenue by at least $20\%$ compared to the previous year?

$(3)$ Suppose $k=0.8a$, what is the water price at which the water department's revenue is minimized for this year? Also, calculate the minimum revenue.

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