
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 currently has 360 kg of type A raw material and 290 kg of type B raw material, planning to use these two types of raw materials to produce 50 pieces of products A and B in total. It is known that producing one piece of product A requires 9 kg of type A raw material and 3 kg of type B raw material, which can yield a profit of 700 yuan, while producing one piece of product B requires 4 kg of type A raw material and 10 kg of type B raw material, yielding a profit of 1200 yuan. Now, let's assume x pieces of product A are produced.  
(1) Please express the total amount of type A and type B raw materials needed to produce products A and B in terms of x.  
(2) Based on the available raw materials, please design a production plan for the number of products A and B.  
(3) If producing one piece of product A can yield a profit of 700 yuan, and producing one piece of product B can yield a profit of 1200 yuan, resulting in a total profit of y yuan for producing both products, write down the function relationship between y and x.  
(4) Combining (2) and (3), calculate which production plan yields the maximum profit, and the maximum profit is.

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