
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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A certain water plant's reservoir has 400 tons of water at the beginning of each day at midnight. While supplying water to residents, it also pumps water into the reservoir at a rate of 60 tons per hour. The total amount of water supplied to residents within $t$ hours is $120\sqrt{6t}$ $(0 \leqslant t \leqslant 24)$.

$(1)$ At what time of the day is the amount of water in the reservoir at its minimum?

$(2)$ If the amount of water in the reservoir is no more than 80 tons, water supply tension occurs. How many hours in a day does this tension occur?

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