
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
Two nonhorizontal, nonvertical lines in the $xy$-coordinate plane intersect to form a $30^{\circ}$ angle. The slope of one line is $4$ times the slope of the other line. What is the greatest possible value of the product of the slopes of these two lines?
A) $\frac{38 - 6\sqrt{33}}{16}$
B) $\frac{38 + 6\sqrt{33}}{16}$
C) $\frac{38}{16}$
D) $\frac{6\sqrt{33}}{16}$

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