
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
Given that points \( B \) and \( C \) are in the fourth and first quadrants respectively, and both lie on the parabola \( y^2 = 2px \) where \( p > 0 \). Let \( O \) be the origin, and \(\angle OBC = 30^\circ\) and \(\angle BOC = 60^\circ\). If \( k \) is the slope of line \( OC \), find the value of \( k^3 + 2k \).

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