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[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

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In $2020$, a great historical achievement was made in the comprehensive construction of a moderately prosperous society, and a decisive victory was achieved in the battle against poverty. A certain city actively explored regional characteristic economies, guided businesses to use the advantages of multimedia to advertise local specialties, and achieved a double harvest of social and economic benefits. A certain business statistically collected data on monthly advertising investment $x$ (in units of ten thousand yuan) and monthly sales volume $y$ (in units of ten thousand pieces) for $7$ months as shown in the table below:

| Monthly Advertising Investment $x$ (ten thousand yuan) | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ |
|--------------------------------------------------------|-----|-----|-----|-----|-----|-----|-----|
| Monthly Sales Volume $y$ (ten thousand pieces)          | $28$| $32$| $35$| $45$| $49$| $52$| $60$|

$(1)$ Given that a linear regression model can be used to fit the relationship between $y$ and $x$, please explain using the correlation coefficient.

$(2)$ Find the linear regression equation for $y$ with respect to $x$, and predict the amount of monthly advertising investment when the monthly sales volume can exceed $70,000 pieces.

Reference data: $\sum_{i=1}^7{({{x_i}-\overline{x}})({{y_i}-\overline{y}})}=150$, $\sum_{i=1}^7{{{({{y_i}-\overline{y}})}^2}=820$, $\sqrt{1435}\approx 37.88$.

Reference formulas: Correlation coefficient $r=\frac{{\sum_{i=1}^n{({{x_i}-\overline{x}})({{y_i}-\overline{y}})}}}{{\sqrt{\sum_{i=1}^n{{{({{x_i}-\overline{x}})}^2}}\sum_{i=1}^n{{{({{y_i}-\overline{y}})}^2}}}}}$; Regression line $\hat{y}=\hat{b}x+\hat{a}$, where the least squares estimates of the slope and intercept are $\hat{b}=\frac{{\sum_{i=1}^n{({{x_i}-\overline{x}})({{y_i}-\overline{y}})}}}{{\sum_{i=1}^n{{{({{x_i}-\overline{x}})}^2}}}}$.

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