
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
Let \( P_a \) be the parabola \( y = \frac{a^3 x^2}{3} + \frac{a^2 x}{2} - 2a \). Find the locus of the vertices of \( P_a \), and the envelope of \( P_a \). Sketch the envelope and two \( P_a \).

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