
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 supermarket plans to distribute 100 boxes of agricultural products of two new brands, $A$ and $B$. The cost price and selling price of these two agricultural products are shown in the table below.

|          | $A$ brand | $B$ brand |
|----------|-----------|-----------|
| Cost Price (in yuan per box) | $80$ | $130$ |
| Selling Price (in yuan per box) | $120$ | $200$ |

$(1)$ If the supermarket spends a total of $10,000$ yuan on these two new brands of agricultural products, how many boxes of each brand should be purchased?

$(2)$ Assuming that the profit from selling each brand of agricultural product remains the same, if the total profit from selling this batch of agricultural products at the supermarket is not less than $5,600$ yuan, how many boxes of brand $B$ should be purchased at least?

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