
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
Suppose a tourist attraction has a fixed cost of $500$ yuan per day, and each ticket is priced at $30$ yuan. The variable cost is directly proportional to the square root of the number of people buying tickets to enter the tourist attraction. When the daily ticket sales reach $25$, the tourist attraction breaks even; when the daily ticket sales exceed $100$ (excluding $100$), the tourist attraction must pay an additional insurance fee of $200$ yuan. Let the daily number of ticket buyers be $x$, and the profit be $y$ yuan.  
$(1)$ Find the function relationship between $y$ and $x$;  
$(2)$ The tourist attraction hopes to avoid losses when the number of visitors reaches $20$. If the measure taken is to increase the ticket price, what is the minimum amount (rounded to the nearest integer) that each ticket must be increased by? (Reference data: $\sqrt {2} \approx 1.41$, $\sqrt {3} \approx 1.73$, $\sqrt {5} \approx 2.24$)

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