
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 response to the call for energy conservation and emission reduction, and to promote a green lifestyle, a certain brand's $4S$ dealership is preparing to purchase a total of $20$ electric cars of two different models, $A$ and $B$.

|         | Cost Price (in $10,000s $/car) | Selling Price (in $10,000s $/car) |
|---------|---------------------------------|-----------------------------------|
| $A$ type | $16$                            | $16.8$                            |
| $B$ type | $28$                            | $29.4$                            |

$(1)$ In order to ensure that the number of $A$ type electric cars purchased by the $4S$ dealership is at least three times the number of $B$ type electric cars, how many $A$ type cars should be purchased at least?

$(2)$ Under the condition in $(1)$, if all $20$ electric cars are sold, in order to maximize the profit of the $4S$ dealership, how many $A$ type electric cars should be purchased? What is the maximum profit?

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