
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
In recent years, the number of college graduates in China has been increasing year by year. Various provinces and cities have introduced preferential policies to encourage college graduates to start their own businesses, in order to drive employment through entrepreneurship. A city has compiled the number of graduates and the number of graduates who started their own businesses (in thousands) from four universities in the year $2021$, as shown in the table below:

|           | $A$ University | $B$ University | $C$ University | $D$ University |
|-----------|---------------|---------------|---------------|---------------|
| Graduates | $3$           | $4$           | $5$           | $6$           |
| Entrepreneurs | $0.1$       | $0.2$         | $0.4$         | $0.5$         |

$(1)$ Given that $y$ and $x$ have a strong linear relationship, find the linear regression equation of $y$ with respect to $x$, $\hat{y} = \hat{a} + \hat{b}x$;  
$(2)$ Assuming that the city government provides a subsidy of $10,000 yuan per person to college students who choose to start their own businesses.  
($i$) If the number of graduates from University $E$ in the city in $2021$ is $7$ thousand, estimate the total amount of subsidies the city government needs to provide to graduates from University $E$ who choose to start their own businesses based on the conclusion in $(1)$;  
($ii$) If the probabilities of Xiao Ming and Xiao Hong, two graduates from University $A$, choosing to start their own businesses are $p$ and $2p-1$ ($\frac{1}{2} < p < 1$), and the city government expects the total amount of subsidies for Xiao Ming and Xiao Hong's entrepreneurship not to exceed $14,000 yuan, find the range of values for $p$.

Reference formula: The least squares estimation formulas for the slope and intercept in the regression equation $\hat{y} = \hat{a} + \hat{b}x$ are $\hat{b} = \frac{{\sum_{i=1}^n{x_i}{y_i}-n\overline{xy}}}{{\sum_{i=1}^n{x_i^2}-n{\overline{x}}^2}}$ and $\hat{a} = \overline{y}-\hat{b}\overline{x}$.

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