
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
"Eating zongzi during the Dragon Boat Festival" is a traditional Chinese custom. Before the Dragon Boat Festival arrives, a certain supermarket purchases a brand of zongzi. The cost of each box is $40, and it is stipulated that the selling price per box must not be less than $50, with a daily sales volume of no less than 350 boxes. Based on past sales experience, it was found that when the selling price per box is set at $50, the daily sales volume is 500 boxes. For every $1 increase in the selling price per box, the daily sales volume decreases by 10 boxes. Let the selling price per box be $x, and the daily sales volume be $p$.

$(1)$ When $x=60$, $p=$______；

$(2)$ At what price per box should it be set to maximize the daily sales profit $W (in dollars)$? What is the maximum profit?

$(3)$ Xiao Qiang said: "When the daily sales profit is maximized, the daily sales revenue is not maximized." Xiao Hong said: "When the daily sales profit is not less than $8000, the range of selling price $x$ is $60\leqslant x\leqslant 80$." Do you think their statements are correct? If correct, please explain the reason; if incorrect, write down the correct conclusion.

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