
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 school's preparation for resumption of classes during the epidemic, in order to implement the idea that "life is more important than Mount Tai, safety is paramount," the school plans to purchase two types of disinfectants, indoor and outdoor. It is known that the price of each barrel of outdoor disinfectant is $30 more than the price of each barrel of indoor disinfectant. Buying $2$ barrels of indoor disinfectant and $3$ barrels of outdoor disinfectant costs a total of $340$ yuan.  
$(1)$ Find the price of each barrel of indoor and outdoor disinfectant.  
$(2)$ According to the actual situation of the school, it is necessary to purchase a total of $200$ barrels of indoor and outdoor disinfectants, and the total cost should not exceed $14,000 yuan. How many barrels of indoor disinfectant should be purchased at least?

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