
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
With the popularization of the concept of "low-carbon living, green travel," new energy vehicles are gradually becoming people's favorite means of transportation. A certain car sales company plans to purchase a batch of two models, $A$ and $B$, of new energy vehicles for sale, with factory prices and selling prices as shown in the table below:

| Model | $A$ | $B$ |
| --- | --- | --- |
| Factory Price | $10$ million yuan per vehicle | $18$ million yuan per vehicle |
| Selling Price | $12$ million yuan per vehicle | $22$ million yuan per vehicle |

The car sales company purchased a total of $130$ vehicles of models $A$ and $B$ last month, with $x$ vehicles of model $A$ purchased. The total profit from selling this batch of new energy vehicles is $y$ million yuan. (Note: Profit $=$ Selling Price $-$ Factory Price)

$(1)$ Find the functional expression between $y$ and $x$ (no need to write the range of the independent variable $x$).

$(2)$ Due to restrictions from the manufacturer and funds, if the number of model $B$ new energy vehicles purchased by the car sales company last month does not exceed twice the number of model $A$ new energy vehicles purchased, please help the car sales company design a purchasing plan to maximize the total profit $y$, and find the maximum value of $y$.

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