
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 gift shop sells two types of gift boxes, $A$ and $B$. In the first purchase, they bought 10 boxes of type $A$ and 15 boxes of type $B$ for a total of 2800 yuan. In the second purchase, they bought 6 boxes of type $A$ and 5 boxes of type $B$ for a total of 1200 yuan. <br/>$(1)$ Find the unit prices of type $A$ and type $B$ gift boxes in yuan. <br/>$(2)$ If the gift shop plans to purchase a total of 40 boxes of the two types of gift boxes for a total cost not exceeding 4500 yuan, how many boxes of type $A$ should they purchase at least?

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