
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
The cost price of a certain product is $20$ yuan. Market research shows that when the selling price is $30$ yuan, 100 units can be sold per week. For every increase of $1$ yuan in price, 2 fewer units are sold per week. Now it is required to sell at least 70 units per week, and the selling price should not be less than $35$ yuan.  
$(1)$ Let the selling price be $x$ yuan and the weekly profit be $y$ yuan. Find the functional relationship between $y$ and $x$, and directly write down the range of values for $x$;  
$(2)$ At what price is the profit maximized for selling this product per week? What is the maximum profit?  
$(3)$ If the weekly profit is not less than $1600$ yuan, find the range of values for the selling price $x$.

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