
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]


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Wearing a safety helmet during cycling can protect the head and reduce injuries. A certain store sells two types of safety helmets, A and B. The table below shows the sales situation in the past two days:

| Time      | Sales of Helmet A | Sales of Helmet B | Sales Amount (元) |
|-----------|-------------------|-------------------|-------------------|
| Day 1     | 10                | 15                | 1150              |
| Day 2     | 6                 | 12                | 810               |

$(1)$ Find the unit prices of helmets A and B.
$(2)$ If the cost prices of helmets A and B are $40$元 per unit and $30$元 per unit, respectively, the store is preparing to purchase a total of $100$ helmets of these two types using funds not exceeding $3400$元.
   ① What is the maximum number of helmets of type A that can be purchased?
   ② Can the store achieve a profit goal of $1300$元 after selling these $100$ helmets? If yes, provide the corresponding procurement plan; if not, explain the reason. (Note: Profit = Selling Price - Cost Price, and the cost price and selling price remain unchanged.)

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