
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
A certain local government is vigorously cultivating and developing the market for elderly care services to relieve the dilemma of empty-nest elderly people lacking daily care and social support. Community elderly care institutions have been newly built since 2016. The table below shows the number of new community elderly care institutions built in the past five years in that area:

| Year | 2016 | 2017 | 2018 | 2019 | 2020 |
|------|------|------|------|------|------|
| Year Code ($x$) | 1 | 2 | 3 | 4 | 5 |
| New Institutions ($y$) | 12 | 15 | 20 | 25 | 28 |

$(1)$ Based on the data in the table, it is known that there is a linear relationship between $y$ and $x$. Use the method of least squares to find the empirical regression equation of $y$ on $x$, $\hat{y} = \hat{b}x + \hat{a}$.

$(2)$ If the elderly people participating in community elderly care in that area have ages $X$ that approximately follow $N(70,9)$, where the age $X$ is in the interval $(76,79]$, and there are 321 people in this age group, estimate how many elderly people are participating in community elderly care in that area?

Reference formulas: Linear regression equation $\hat{y} = \hat{b}x + \hat{a}$, $\hat{b} = \frac{{\sum_{i=1}^n{({{x_i}-\overline{x}})}({{y_i}-\overline{y}})}}{{\sum_{i=1}^n{{{({{x_i}-\overline{x}})}^2}}}}$, $\hat{a} = \overline{y} - \hat{b}\overline{x}$.

Reference data: $P(\mu -2\sigma \leq X \leq \mu +2\sigma) = 0.9545$, $P(\mu -3\sigma \leq X \leq \mu +3\sigma) = 0.9973$.

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