
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 $f(x) = x(e^x - 1) - ax^2$, where $e$ is the base of the natural logarithm.

1. If $a = \frac{1}{2}$, find the monotonic intervals of $f(x)$.
2. If $f(x) \geq 0$ when $x \geq 0$, find the range of values for $a$.
3. If $f(x)$ has no extreme values, find the value of $a$.

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