
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 that was continuously dripping water due to damage. In order to investigate the waste caused by the water leakage, Xiao Ming placed a graduated cylinder under the faucet to collect water and recorded the total amount of water in the cylinder every minute. However, 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) | $1$ | $2$ | $3$ | $4$ | $5$ | $\ldots$ |
|--------------------|-----|-----|-----|-----|-----|---------|
| Total water amount $y$ (milliliters) | $7$ | $12$ | $17$ | $22$ | $27$ | $\ldots$ |

$(1)$ Investigation: Based on the data in the table above, determine the function relationship between the total water amount $y$ and time $t$. Find the expression of $y$ in terms of $t$.

$(2)$ Application:
1. Estimate the total water amount in the cylinder when Xiao Ming measures it at the $20$th 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 (assuming $30$ days) can supply drinking water for one person.

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