
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
During the epidemic prevention and control period, a certain middle school purchased a batch of hand sanitizer and $84$ disinfectant to ensure the health and safety of teachers and students. If they buy $100$ bottles of hand sanitizer and $150$ bottles of $84$ disinfectant, the total cost is $1500$ yuan; if they buy $120$ bottles of hand sanitizer and $160$ bottles of $84$ disinfectant, the total cost is $1720$ yuan.  
$(1)$ What is the price per bottle of hand sanitizer and per bottle of $84$ disinfectant?  
$(2)$ A certain pharmacy sells hand sanitizer, and for every purchase of $150$ bottles of hand sanitizer, they give away $10$ bottles of $84$ disinfectant for free. If the school purchases $150$ bottles of hand sanitizer and $60$ bottles of $84$ disinfectant from this pharmacy, how much will it cost in total?

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