
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 $f\left(x\right)=ax^{3}+2\sin x-x\cos x$.  
$(1)$ If $a=0$, determine the monotonicity of $f\left(x\right)$ on $(-\frac{π}{2},\frac{π}{2})$, and explain the reason.  
$(2)$ When $a \gt 0$, investigate the number of extreme points of $f\left(x\right)$ on $\left(0,\pi \right)$.

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