
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 recent years, the awareness of traffic safety among citizens has gradually increased, leading to a greater demand for helmets. A certain store purchased two types of helmets, type A and type B. It is known that they bought 20 type A helmets and 30 type B helmets, spending a total of 2920 yuan. The unit price of type A helmets is 11 yuan higher than the unit price of type B helmets.  
$(1)$ What are the unit prices of type A and type B helmets, respectively?  
$(2)$ The store decides to purchase 40 more helmets of type A and type B. They coincide with a promotional event by the manufacturer: type A helmets are sold at 20% off the unit price, and type B helmets are discounted by 6 yuan each. If the number of type A helmets purchased this time is not less than half the number of type B helmets, how many type A helmets should be purchased to minimize the total cost of this purchase? What is the minimum cost?

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