
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
Consider two lines \( L_1 \) and \( L_2 \) with equations \( y = px \) and \( y = qx \) respectively. It is given that \( L_1 \) makes three times the angle with the horizontal x-axis as \( L_2 \) does and that \( L_1 \) has a negative slope that is half the magnitude of the slope of \( L_2 \). If \( q \) is positive and both lines are not vertical, determine the value of \( pq \).

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