
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
"4G changes life, 5G changes society." Different types of 5G phones bring people a whole new experience. A certain business hall currently sells two models of 5G phones, denoted as $A$ and $B$. It is known that selling 1 unit of type $A$ and 1 unit of type $B$ phones together yields a profit of $600$ yuan, while selling 3 units of type $A$ and 2 units of type $B$ phones together yields a profit of $1400$ yuan. <br/>$(1)$ Find the profit per unit for each type of phone, $A$ and $B$. <br/>$(2)$ The business hall purchases a total of 20 units of type $A$ and $B$ phones. The number of type $B$ phones purchased does not exceed $\frac{2}{3}$ of the number of type $A$ phones. Design a purchasing plan to maximize the profit when selling these 20 units of phones, and calculate the maximum profit.

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