
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 $f(x) = e^{x} - \sin x - \cos x$, where $f'(x)$ is its derivative.  
$(1)$ Find the number of extreme points of $f(x)$ on the interval $[-\pi, +\infty)$.  
$(2)$ If $f'(x) \geq ax + 2 - 2\cos x$ holds for all $x \in [-\pi, +\infty)$, find the value of $a$.

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