
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
A health and medical equipment factory needs to produce a batch of high-quality medical masks within 8 days (including the 8th day), totaling 50,000 masks of type A and type B, with no fewer than 18,000 of type A masks. The factory's production capacity is: only one type of mask can be produced per day. If producing type A masks, 6,000 can be produced per day; if producing type B masks, 8,000 can be produced per day. The profit from producing one type A mask is 0.5 yuan, and from one type B mask is 0.3 yuan. Assuming the factory produces $x$ ten thousand type A masks, the questions are:  
① How much profit in ten thousand yuan can the factory make from producing type A masks? And how much from type B masks?  
② If the total profit from producing masks is $y$ ten thousand yuan, find the function of $y$ in terms of $x$ and give the range of values for the variable $x$;  
③ If you were the factory manager, under the premise of completing the task, how would you arrange the production of type A and type B masks to maximize the total profit? What is the maximum profit? If the requirement is to complete the task in the shortest time, how would you arrange the production of type A and type B masks? What is the shortest time?

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