
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]


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During the epidemic, a company ordered $A$ and $B$ two types of hand sanitizers totaling 50 boxes from the manufacturer. It is known that the purchase price of each box of type $A$ hand sanitizer is $200 yuan. On this basis, for each additional box of type $A$ hand sanitizer purchased, the purchase price decreases by $2 yuan per box. The purchase price of type $B$ hand sanitizer is $100 yuan per box. Both types $A$ and $B$ of hand sanitizers must not be less than 15 boxes. Let the company purchase $x$ boxes of type $A$ hand sanitizer.<br/>$(1)$ Fill in the table according to the information;<br/>$(2)$ If the company wants to minimize the cost, how should they design the ordering plan?<br/><table><tbody><tr><td width="93">Type</td><td width="215">Quantity (boxes)</td><td width="262">Purchase Price (yuan/box)</td></tr><tr><td>$A$</td><td>$x$</td><td>______</td></tr><tr><td>$B$</td><td>______</td><td>$100$</td></tr></tbody></table>

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