
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]


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A construction company in Changsha has a total of 50 slag transport vehicles, including 20 Type A and 30 Type B vehicles. Now all these slag transport vehicles are allocated to the construction of the Changsha-Zhuzhou-Xiangtan intercity light rail, with 30 vehicles sent to Site A and 20 vehicles sent to Site B. The daily rental prices agreed upon by the two sites and the construction company are as follows:

|            | Type A Slag Truck Rental | Type B Slag Truck Rental |
|------------|--------------------------|--------------------------|
| Site A     | $1800 per vehicle        | $1600 per vehicle        |
| Site B     | $1600 per vehicle        | $1200 per vehicle        |

$(1)$ If $x$ Type A slag transport vehicles are sent to Site A, the rental income obtained by the construction company for these 50 slag transport vehicles in one day is $y (in yuan)$. Find the analytical expression of $y$ in terms of $x$.

$(2)$ If the total rental income of these 50 slag transport vehicles in one day is not less than $79600 yuan$, determine how many allocation schemes exist and list all the schemes.

$(3)$ Under the condition of $(2)$, which allocation scheme will result in the construction company earning the highest daily rental income? What is the maximum rental income? Please explain the reason.

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