
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
A certain car trading company sells two models of new energy cars, $A$ and $B$. The purchase price of model $A$ is $120,000$ yuan per unit, and the purchase price of model $B$ is $150,000$ yuan per unit. The company can make a profit of $31,000$ yuan by selling $2$ units of model $A$ and $5$ units of model $B$, and a profit of $13,000$ yuan by selling $1$ unit of model $A$ and $2$ units of model $B$.  
$(1)$ Find the profit of selling one unit of model $A$ and one unit of model $B.  
$(2)$ The company plans to use no more than $3$ million yuan to purchase a total of $22$ units of new energy cars of models $A$ and $B$. How many units of model $A$ should be purchased at least?

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