
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 certain shopping mall in our city has two types of goods, A and B. The cost price of type A is $15$ yuan per unit, and the selling price is $20$ yuan; the cost price of type B is $35$ yuan per unit, and the selling price is $45$ yuan. The merchant purchases a total of $100$ units of type A and type B goods. Let $x$ represent the number of units of type A goods purchased. The total profit after selling these two types of goods is $y$ yuan.

(1) Write the functional relationship between $y$ and $x$.

(2) The merchant plans to invest a maximum of $3000$ yuan to purchase these two types of goods. How many units of type A goods should be purchased at least? What is the maximum profit the merchant can make after selling these goods?

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