
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 movie theater has a total of $1000$ seats, with no distinction in ticket prices. According to the theater's operating experience, when the ticket price does not exceed $10$ yuan, all tickets can be sold out. When the ticket price is higher than $10$ yuan, for each increase of $1$ yuan, $30$ tickets cannot be sold. To achieve better revenue, an appropriate ticket price needs to be set for the theater, with the following basic conditions: $①$ For the convenience of making change and accounting, the ticket price is set as an integer multiple of $1$ yuan; $②$ The cost of showing a movie in the theater is $5750$ yuan, and the box office revenue must be higher than the cost. Using $x(\text{yuan})$ to represent the ticket price and $y(\text{yuan})$ to represent the net revenue of the theater (revenue after deducting the cost),

$(1)$ Express $y$ as a function of $x$ and find its domain;
$(2)$ Given the basic conditions, what ticket price yields the maximum net revenue per screening?

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