
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
Go, originated in China, was called "Yi" in ancient times, and is the ancestor of board games, with a history of more than 4000 years. A certain merchant sells two types of Go boards, $A$ and $B$, with purchase prices of $200$ yuan and $170$ yuan respectively. The table below shows the sales situation in the past two months:<br/>

| Sales Period | Quantity of Type A | Quantity of Type B | Sales Revenue |
|--------------|--------------------|--------------------|---------------|
| First month  | 3 sets             | 5 sets             | 1800 yuan     |
| Second month | 4 sets             | 10 sets            | 3100 yuan     |

$(1)$ Find the selling price per set for types $A$ and $B$ of Go boards.<br/>
$(2)$ If the merchant is prepared to spend no more than 5400 yuan to purchase a total of 30 sets of types $A$ and $B$ of Go boards, find the maximum number of sets of type $A$ that can be purchased.<br/>
$(3)$ Under the condition in $(2)$, can the store achieve a profit goal of 1300 yuan after selling these 30 sets of Go boards? Please explain.

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