
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
Market researchers of an electronics company conducted a statistical analysis of the market share of the company over the past seven months, with the results shown in the table below:

| Month | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ |
|-------|-----|-----|-----|-----|-----|-----|-----|
| Month Code $x$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ |
| Market Share $y(\%)$ | $11$ | $13$ | $16$ | $15$ | $20$ | $21$ | $23$ |

$(1)$ Use the correlation coefficient to determine if the relationship between market share $y$ and month code $x$ can be modeled by a linear regression model. (Keep the result to two decimal places)<br/>$(2)$ Find the linear regression equation for $y$ in terms of $x$ and predict the market share for the company in October. <br/>Reference: $\sum_{i=1}^{7}({x}_{i}-\overline{x})^{2}=28$, $\sum_{i=1}^{7}({y}_{i}-\overline{y})^{2}=118$, $\sum_{i=1}^{7}({x}_{i}-\overline{x})({y}_{i}-\overline{y})=56$, $\sqrt{826}≈28.7$.<br/>Reference formulas: Correlation coefficient $r=\frac{\sum_{i=1}^{n}({x}_{i}-\overline{x})({y}_{i}-\overline{y})}{\sqrt{\sum_{i=1}^{n}({x}_{i}-\overline{x})^{2}}\sqrt{\sum_{i=1}^{n}({y}_{i}-\overline{y})^{2}}}$, Linear regression equation $\hat{y}=\hat{b}x+\hat{a}$ with the least squares estimates of slope and intercept as $\hat{b}=\frac{\sum_{i=1}^{n}({x}_{i}-\overline{x})({y}_{i}-\overline{y})}{\sum_{i=1}^{n}({x}_{i}-\overline{x})^{2}}$, $\hat{a}=\overline{y}-\hat{b}\overline{x}$.

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