
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
To combat the "novel coronavirus pneumonia," a medical equipment factory in our city has accepted the task of producing a batch of high-quality medical masks. The task requires producing a total of 2 million masks of type $A$ and type $B$ within 30 days (including 30 days). Due to certain constraints in actual production, the factory can only produce one type of mask per day. It is known that the factory can produce twice as many type $A$ masks per day as type $B$ masks. Additionally, producing 40 million type $A$ masks takes 6 days less than producing 50 million type $B$ masks.

$(1)$ How many million type $B$ masks can the factory produce per day?

$(2)$ If the profit from producing one type $A$ mask is $0.8$ yuan and the profit from producing one type $B$ mask is $1.2$ yuan, how should the factory schedule production to maximize profit while ensuring timely delivery of this batch of masks?

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