
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
Xiao Ming observed a faucet continuously dripping water due to damage. To investigate the waste caused by the water leakage, Xiao Ming placed a graduated cylinder under the faucet to collect water. He recorded the total amount of water in the cylinder every minute, but due to a delay in starting the timer, there was already a small amount of water in the cylinder at the beginning. Therefore, he obtained a set of data as shown in the table below:

| Time $t$ (minutes) | 1 | 2 | 3 | 4 | 5 | ... |
|---------------------|---|---|---|---|---|----|
| Total water amount $y$ (milliliters) | 7 | 12 | 17 | 22 | 27 | ... |

$(1)$ Investigation: Based on the data in the table above, determine which one of the functions $y=\frac{k}{t}$ and $y=kt+b$ (where $k$ and $b$ are constants) can correctly reflect the functional relationship between the total water amount $y$ and time $t$. Find the expression of $y$ in terms of $t$.

$(2)$ Application:

① Estimate how many milliliters of water will be in the cylinder when Xiao Ming measures it at the 20th minute.

② A person drinks approximately 1500 milliliters of water per day. Estimate how many days the water leaked from this faucet in a month (30 days) can supply one person.

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