
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 a function $f(x) = xe^{1-x} + ax^{3} - x$ where $a \in \mathbb{R}$.<br/>$(1)$ If $a = \frac{1}{3}$, determine the monotonicity of $f(x)$ on $(-\infty, 0]$;<br/>$(2)$ If the inequality $f(x) \geq x\ln x + a$ holds for all $x \in [1, +\infty)$, find the range of values for $a$.

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