
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, non vertical lines in the $xy$-coordinate plane intersect to form a $30^{\circ}$ angle. One line has a slope that is $4$ times the slope of the other line. What is the greatest possible value of the product of the slopes of the two lines?
A) \(\frac{1}{\sqrt{3}}\)
B) \(\frac{3\sqrt{3} + \sqrt{11}}{16}\)
C) \(\frac{(3\sqrt{3} + \sqrt{11})^2}{16}\)
D) \(\frac{3\sqrt{3} - \sqrt{11}}{16}\)
E) \(\frac{(3\sqrt{3} - \sqrt{11})^2}{16}\)

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