
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 the function f(x) = xlnx + $\frac{1}{6}$x^3 + ax^2 - x + 1.
1. If the derivative of the function f(x), g(x), is monotonically increasing in its domain, find the range of values for the real number a.
2. When a = $- \frac{1}{8}$, the function h(x) = f(x) - $\frac{1}{6}$x^3 has two extreme points x1 and x2 in its domain, with x1 < x2. Compare the size of x1 \* x2^2 and e^3, and explain the reason.

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