
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
The average deposit per capita $y$ of households in a certain region from $2015$ to $2021$ is shown in the table below:

| Year | $2015$ | $2016$ | $2017$ | $2018$ | $2019$ | $2020$ | $2021$ |
|------|--------|--------|--------|--------|--------|--------|--------|
| Year Code $t$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ |
| Average Deposit $y$ | $2.9$ | $3.3$ | $3.6$ | $4.4$ | $4.8$ | $5.2$ | $5.9$ |

The variables $t$ and $y$ have a linear relationship. Three students, A, B, and C, calculated the regression line equations as follows: A $y=0.5t-2.3$; B $y=-0.5t+2.3$; C $y=0.5t+2.3$. Only one student's calculation is correct.

$(1)$ Determine whose calculation result is correct.

$(2)$ If the error between the estimated data obtained from the linear regression equation and the test data is greater than $0.1$, the data is considered "unreliable." If the error is $0$, the test data is considered "perfect." Any data other than these two cases is considered "reliable." Remove the unreliable data, randomly select $2$ from the remaining data, and find the distribution table and mathematical expectation of the number of "perfect" data among them.

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