
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]


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During an underwater archaeological activity, a diver needs to dive $50$ meters to the bottom of the water for archaeological work. The oxygen consumption consists of the following three aspects: $(1)$ The average diving speed is $x$ meters/minute, and the oxygen consumption per minute is $\frac{1}{100}x^{2}$ liters; $(2)$ The working time at the bottom of the water is between $10$ and $20$ minutes, and the oxygen consumption per minute is $0.3$ liters; $(3)$ When returning to the water surface, the average speed is $\frac{1}{2}x$ meters/minute, and the oxygen consumption per minute is $0.32$ liters. The total oxygen consumption of the diver in this archaeological activity is $y$ liters.
$(1)$ If the working time at the bottom of the water is $10$ minutes, express $y$ as a function of $x$;
$(2)$ If $x \in [6, 10]$, the working time at the bottom of the water is $20$ minutes, find the range of total oxygen consumption $y$;
$(3)$ If the diver carries $13.5$ liters of oxygen, how many minutes can the diver stay underwater at most (round to the nearest whole number)?

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