
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), requiring a total of 50,000 masks of two types, Type A and Type B, with no fewer than 18,000 of Type A. The factory's production capacity is: only one type of mask can be produced per day, with 6,000 of Type A or 8,000 of Type B produced daily. 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, answer the following questions:  
① What is the profit from producing Type A masks in ten thousand yuan? What is the profit from producing Type B masks in ten thousand yuan?  
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 x;  
③ If you were the factory manager and wanted to maximize profit while completing the task, how would you arrange the production of Type A and Type B masks? What is the maximum profit? If the goal is to complete the task in the shortest possible time, how would you arrange the production? What is the shortest time?

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