
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
A certain distributor ordered two popular types of rice dumplings from the manufacturer, and the purchase prices and selling prices of the two types of rice dumplings are shown in the table below:

| Category | Type $A$ | Type $B$ |
|----------|----------|----------|
| Purchase Price (元/box) | $25$ | $30$ |
| Selling Price (元/box) | $32$ | $40$ |

$(1)$ If the distributor spends $1500$ yuan to purchase types $A$ and $B$ of rice dumplings, with the quantity of type $A$ being $4$ boxes less than twice the quantity of type $B$, find out how many boxes of type $A$ and type $B$ were purchased respectively.

$(2)$ If the distributor plans to purchase at least twice as many boxes of type $A$ rice dumplings as type $B$ rice dumplings, and plans to purchase a total of $60$ boxes of the two types of rice dumplings, how should the distributor design the purchasing plan to maximize the profit after selling? What is the maximum profit?

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