
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
"The epidemic is a command, and prevention and control is a responsibility." A certain company in Nancheng County, Fuzhou City, plans to purchase a certain quantity of disinfectants from brand A and brand B at the same time for epidemic prevention and resumption of work. If they buy 20 bottles of brand A disinfectant and 10 bottles of brand B disinfectant, the total cost is 1300 yuan; if they buy 10 bottles of brand A disinfectant and 10 bottles of brand B disinfectant, the total cost is 800 yuan.<br/>$(1)$ What is the unit price of brand A and brand B disinfectants, respectively?<br/>$(2)$ The company plans to purchase a total of 50 bottles of brand A and brand B disinfectants, with a budget not exceeding 1900 yuan for both products. They also require the quantity of brand A disinfectant to be no less than half of the quantity of brand B disinfectant. How many purchasing plans does the company have?

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