
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
Livestreaming e-commerce is boosting rural revitalization, as a novel way to sell local specialties, it is highly sought after by all sectors of society. A certain livestreaming room conducted a live broadcast activity for local landmark products, and the data statistics of the number of live broadcast cycles $x$ (where $10$ sessions make up one cycle) and product sales $y$ (in thousands of yuan) are as follows:

| Number of Live Broadcast Cycles $x$ | 1 | 2 | 3 | 4 | 5 |
| --- | --- | --- | --- | --- | --- |
| Product Sales $y$ (in thousands of yuan) | 3 | 7 | 15 | 30 | 40 |

Based on the data characteristics, Person A believes that the sample points are distributed around an exponential curve $y=2^{bx+a}$, and based on this, he performed some preliminary data processing. The table below shows the results:

| $\overline{z}$ | $\sum_{i=1}^5{x_i^2}$ | $\sum_{i=1}^5{{x_i}{y_i}}$ | $\sum_{i=1}^5{{x_i}{z_i}}$ | $\sum_{i=1}^5{{{({{y_i}-\overline{y}})}^2}}$ | $\sum_{i=1}^5{{{({{y_i}-{{\hat{y}}_i}})}^2}}$ |
| --- | --- | --- | --- | --- | --- |
| 3.7 | 55 | 382 | 65 | 978 | 101 |

Where $z_{i}=\log _{2}y_{i}$, $\overline{z}=\frac{1}{5}\sum_{i=1}^{5}{z}_{i}$.

(1) Based on the data in the table, establish the regression equation of $y$ with respect to $x$ (coefficients accurate to $0.01$).

(2) 
- Person B believes that the sample points are distributed around the line $y=mx+n$, and calculated the regression equation to be $\hat{y}=9.7x-10.1$, with a corresponding coefficient of determination $R_{B}^{2}=0.98$. Compare the models established by Person A and Person B, and determine whose fitting effect is better.
- Based on the conclusion from (1), calculate the minimum number of live broadcast cycles required for the sales to exceed 80,000 yuan. (Round the final answer to the nearest whole number).

Given: For a set of data $(u_{1},v_{1}), (u_{2},v_{2}), \ldots, (u_{n},v_{n})$, the least squares estimates of the slope and intercept of the regression line $\hat{v}=\hat{\alpha}+\hat{\beta}u$ are $\hat{\beta}=\frac{\sum_{i=1}^{n}(u_{i}-\overline{u})(v_{i}-\overline{v})}{\sum_{i=1}^{n}(u_{i}-\overline{u})^{2}}$, $\hat{\alpha}=\overline{v}-\hat{\beta}\overline{u}$, and the coefficient of determination is $R^{2}=1-\frac{\sum_{i=1}^{n}(v_{i}-\widehat{v_{i}})^{2}}{\sum_{i=1}^{n}(v_{i}-\overline{v})^{2}}$.

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