
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 factory has 66 kg of raw material A and 66.4 kg of raw material B. It plans to produce two types of products, A and B, totaling 90 pieces. It is known that each piece of product A requires 0.5 kg of raw material A and 0.8 kg of raw material B, while each piece of product B requires 1.2 kg of raw material A and 0.6 kg of raw material B.  
$(1)$ What are the possible production plans for the factory?  
$(2)$ If all the products are sold, and the profit for each piece of product A is $30, and for each piece of product B is $20, which plan from $(1)$ will yield the highest profit? What is the maximum profit?

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