
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 have become the preferred mode of transportation for most people. The Lighthouse City Bus Company purchased a batch of two models, $A$ and $B$, of new energy vehicles. It is known that purchasing $3$ $A$-type cars and $1$ $B$-type car requires a total of $55$ million yuan, and purchasing $2$ $A$-type cars and $4$ $B$-type cars requires a total of $120$ million yuan.  
$(1)$ Find out how much each $A$-type and $B$-type car costs.  
$(2)$ If the company plans to purchase a total of $15$ cars of $A$ and $B$ types, and the total cost does not exceed $220$ million yuan, then how many $A$-type cars can be purchased at least?

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