
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
Xiao Li sells a kind of souvenir in a scenic spot. It is known that the cost price of each item is $6$ yuan. When the selling price is set at $8$ yuan, he can sell $200$ items per day. Market research shows that for every $1$ yuan increase in the selling price, the daily sales volume will decrease by $10$ items. The Department of Commerce stipulates that the selling price cannot exceed $12$ yuan. Let the selling price of the souvenir be $x$ (yuan), the daily sales volume be $y$ (items), and the daily sales profit be $w$ (yuan).  
$(1)$ Find the functional relationship between $y$ and $x$.  
$(2)$ To make the daily sales profit $720$ yuan, what should be the selling price?  
$(3)$ Find the functional relationship between the daily sales profit $w$ (yuan) and the selling price $x$ (yuan). For what value of $x$ is the daily sales profit maximized, and what is the maximum profit?

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