
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 $f(x)=(x-1)e^x-ax^2$ (where $e$ is the base of the natural logarithm, $a\in \mathbb{R}$),

$(1)$ Determine the number of extreme points of the function $f(x)$, and explain the reason;

$(2)$ If the inequality $f(x)+e^x\geqslant x^3+x$ holds for $\forall x\in \mathbb{R}$, find the range of values for $a$.

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