
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 various speeds, and the number of defective items it produces per hour varies with the speed of operation. Let $x$ represent the speed (in revolutions per second), and $y$ represent the number of defective items produced per hour. Now, we have observed 4 sets of values for $(x, y)$: $(8, 5)$, $(12, 8)$, $(14, 9)$, and $(16, 11)$.

(Ⅰ) Assuming that there is a linear relationship between $y$ and $x$, find the equation of the regression line between $y$ and $x$.

(Ⅱ) If the maximum number of defective items allowed per hour in actual production is 10, what is the maximum speed (in revolutions per second) the machine can operate at? (Round to the nearest whole number)

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