
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 $f(x) = xe^{x} - ax$ where $a \in \mathbb{R}$.  
$(Ⅰ)$ If $y = f(x)$ is an increasing function on $\mathbb{R}$, find the range of real numbers for $a$.  
$(Ⅱ)$ When $a = 1$, determine if $0$ is a critical point of the function $f(x)$ and explain the reason.  
$(Ⅲ)$ Determine the number of zeros of $f(x)$ and explain the reason.

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