
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
In 2023, "street vendor economy" became a hot topic of social concern. The "street vendor economy" has the advantages of low start-up capital and low management costs. Especially in the economic recovery period after being impacted by the epidemic, the "street vendor economy" has become the first choice for many entrepreneurs. Person A operates a business selling a certain brand of small electrical appliances. Person A needs $90$ yuan to purchase $2$ units of type $A$ small electrical appliances and $3$ units of type $B$ small electrical appliances, and $65$ yuan to purchase $3$ units of type $A$ small electrical appliances and $1$ unit of type $B$ small electrical appliance. Selling one unit of type $A$ small electrical appliance earns a profit of $3$ yuan, and selling one unit of type $B$ small electrical appliance earns a profit of $4$ yuan.<br/>$(1)$ Find out how much it costs to buy $1$ unit of type $A$ small electrical appliance and $1$ unit of type $B$ small electrical appliance each?<br/>$(2)$ Person A wants to purchase a total of $150$ units of type $A$ and type $B$ small electrical appliances with a budget of no less than $2750$ yuan but not exceeding $2850$ yuan. Find the range of values for the number of type $A$ small electrical appliances to be purchased.<br/>$(3)$ Under the conditions of $(2)$, after selling all the purchased type $A$ and type $B$ small electrical appliances, the total profit obtained should be no less than $565 yuan. Please explain what reasonable purchasing plans Person A can have. Calculate which purchasing plan yields the maximum profit, and what is the maximum profit?

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