
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
During the summer vacation of 2022, the outstanding student cadres of the medical school volunteered to conduct nucleic acid testing in the community. Before each nucleic acid test, the volunteers need 10 minutes for preparation. Once the testing starts, the volunteers can test 30 residents per minute. Let $x$ represent the minutes elapsed since the start of the preparation, and $y$ represent the number of residents who have completed nucleic acid testing after $x$ minutes. 

$(1)$ The function relationship between $y$ and $x$ is ______.  
$(2)$ Given that there are a total of 6000 residents in the community, how many minutes (including preparation time) do the volunteers need to complete nucleic acid testing for all residents?

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