
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 machine can operate at different speeds, and some of the products it produces may have defects. The number of defective products produced per hour varies with the speed of the machine. Let $x$ denote the speed of the machine (in revolutions per second), and let $y$ denote the number of defective products produced per hour. Four observed values of $(x, y)$ are $(8,5)$, $(12,8)$, $(14,9)$, and $(16,11)$. It is known that $y$ has a strong linear correlation with $x$. If the actual production process allows for no more than 10 defective products per hour, what should be the maximum speed of the machine in revolutions per second? (Round to the nearest integer)

Reference formula:
If $(x\_1, y\_1), ..., (x\_n, y\_n)$ are sample points, then $\hat{y} = \hat{b}x + \hat{a}$,
$\overline{x} = \frac{1}{n} \sum\_{i=1}^{n} x\_i$, $\overline{y} = \frac{1}{n} \sum\_{i=1}^{n} y\_i$, $\hat{b} = \frac{\sum\_{i=1}^{n} (x\_i - \overline{x}) (y\_i - \overline{y})}{\sum\_{i=1}^{n} (x\_i - \overline{x})^2} = \frac{\sum\_{i=1}^{n} x\_i y\_i - n \overline{x} \overline{y}}{\sum\_{i=1}^{n} x\_i^2 - n \overline{x}^2}$, $\hat{a} = \overline{y} - \hat{b} \overline{x}$.

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