
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 city is hosting a badminton competition for high school students. The total cost $y (\text{in yuan})$ consists of two parts: the venue fee $b (\text{in yuan})$ and the cost of materials, which is directly proportional to the number of participants $x (\text{people})$. The data in the table below was obtained through actual research.

| $x$ | $20$ | $30$ |
|-----|------|------|
| $y$ | $1600$ | $2000$ |

$(1)$ Find the functional relationship between $y$ and $x$;  
$(2)$ If the total cost cannot exceed $3000$ yuan, what is the maximum number of participants in the competition?  
$(3)$ If the number of participants is approximately between $50$ and $70$ people, find the approximate range of the total cost.

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