
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 supermarket sells two types of thermos cups, $A$ and $B$. It is known that the unit price of type $B$ thermos cup is $10$ yuan more than that of type $A$ thermos cup. The quantity of type $B$ thermos cups that can be purchased with $600$ yuan is the same as the quantity of type $A$ thermos cups that can be purchased with $480$ yuan.  
$(1)$ What are the unit prices of thermos cups $A$ and $B$ respectively?  
$(2)$ Due to high demand, the supermarket plans to purchase a total of $120$ thermos cups of types $A$ and $B$, with the quantity of type $A$ thermos cups not less than half of the quantity of type $B$ thermos cups. If the unit price of type $A$ thermos cup remains the same and the unit price of type $B$ thermos cup decreases by $10\%$, and the cost price of each thermos cup is $30$ yuan, how should the purchase be made to maximize the profit from selling these thermos cups, and what is the maximum profit?

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