
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 of a certain component is $30$ yuan per piece. When the selling price is $80$ yuan per piece, the weekly sales volume is 600 pieces. After a market survey, it was found that for every $2$ yuan decrease in the selling price of each component (the decrease in price is an even number), the weekly sales volume would increase by 30 pieces. Let $x$ represent the decrease in price of each component in yuan, and $y$ represent the weekly sales volume when the price decreases by $x$ yuan. Find the functional expression between $y$ and $x$.

$(1)$ Find the functional expression between $y$ and $x$;  
$(2)$ Determine by how much each component should be reduced in price for the weekly sales profit to be maximized, and what is the maximum profit?

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