
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
Yamei Garment Factory currently has 70 meters of Type A fabric and 52 meters of Type B fabric. It plans to use these two types of fabrics to produce a total of 80 sets of Type Jia and Type Yi fashion. It is known that making one set of Type Jia fashion requires 1.1 meters of Type A fabric and 0.4 meters of Type B fabric, yielding a profit of 50 yuan; making one set of Type Yi fashion requires 0.6 meters of Type A fabric and 0.9 meters of Type B fabric, yielding a profit of 45 yuan. Let the number of sets of Type Jia fashion produced be $x$, and the total profit obtained from producing these two types of fashion with this batch of fabric be $y$ yuan.
(1) Find the function relationship between $y$ (yuan) and $x$ (sets),
(2) How many production plans are there?
(3) When the number of sets of Type Jia fashion is how many, can the factory obtain the maximum profit? What is the maximum profit? (Answer using the knowledge of functions learned)

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