
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 $f(x) = e^{ax} - ax$ where $a \in \mathbb{R}$ and $a \neq 0$.
$(1)$ Discuss the monotonicity of $f(x)$.
$(2)$ If the inequality $f(x) \geq \sin x - \cos x + 2 - ax$ holds for all $x \geq 0$, find the range of real number $a$.

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