
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 store introduced a new batch of helmets at a price of $60$ yuan each. After market research, it was found that if the selling price was set at $100$ yuan each, the store could sell 200 helmets per month. In order to cooperate with the traffic management department's campaign of "one belt (safety belt) and one helmet," the store planned to reduce the price of the helmets. After investigation, it was found that for every $4$ yuan reduction in price, an additional 40 helmets could be sold per month. Let $x$ represent the price of each helmet after the reduction, where $60 \leq x \leq 100$, and let $y$ represent the number of helmets sold per month.

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

$(2)$ If the monthly profit from selling helmets at the store is $w$ yuan, find the functional relationship between $w$ and $x$.

$(3)$ Under the condition in $(2)$, at what value of $x$ does the monthly profit $w$ have the maximum value? What is the maximum value?

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