
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 $f(x) = xe^{x} - ax + a$, where $a > 0$.  
$(Ⅰ)$ If $a = 1$, find the monotonic interval of $f(x)$.  
$(Ⅱ)$ If the inequality $f(x) \geq a\ln x$ holds for all $x$, find the range of real numbers for $a$.

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