
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
"To taste a hundred fruits is to become immortal", eating various fruits is beneficial to health. To meet the needs of customers, a fruit shop owner purchases two types of fruits, $A$ and $B$. If $9$ kilograms of type $A$ fruit and $10$ kilograms of type $B$ fruit are purchased, it costs $163$ yuan; if $12$ kilograms of type $A$ fruit and $8$ kilograms of type $B$ fruit are purchased, it costs $164$ yuan.  
$(1)$ What is the price per kilogram of type $A$ and type $B$ fruits respectively?  
$(2)$ According to market rules, the owner needs to purchase fruits in whole kilograms. If selling $1$ kilogram of type $A$ fruit earns a profit of $4$ yuan, and selling $1$ kilogram of type $B$ fruit earns a profit of $6$ yuan, he decides to purchase $A$ fruit in a quantity that is $2$ times more than the quantity of $B$ fruit plus an additional $4$ kilograms. The maximum quantity of type $B$ fruit that can be purchased is $13$ kilograms. After selling all the fruits, the total profit should not be less than $160$ yuan. What are the possible purchasing plans?

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