
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
Under the blue sky and white clouds, among the green mountains and waters, set up a tent, invite family and friends, listen to the cicadas, smell the clear breeze, talk about daily life, how pleasant it is. In response to the "Guiding Opinions on Promoting the Orderly Development of Camping Tourism for Leisure and Health" issued by the Ministry of Culture and Tourism, a certain scenic area needs to purchase two models of tents, $A$ and $B$. If $2$ tents of type $A$ and $4$ tents of type $B$ are purchased, it will cost $5200$ yuan; if $3$ tents of type $A$ and $1$ tent of type $B$ are purchased, it will cost $2800$ yuan. <br/>$(1)$ Find the price of each tent of type $A$ and each tent of type $B$; <br/>$(2)$ If the scenic area needs to purchase a total of $20$ tents of types $A$ and $B$ (both types of tents need to be purchased), and the number of type $A$ tents purchased does not exceed $\frac{1}{3}$ of the number of type $B$ tents purchased, in order to minimize the total cost of purchasing tents, how many tents of type $A$ and type $B$ should be purchased? What is the minimum total cost of purchasing tents?

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