
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 $f(x)=e^{x}-1-a\sin x$ where $a\in \mathbb{R}$.  
$(1)$ If the tangent line to the curve $y=f(x)$ at $(0,f(0))$ is $y=-x$, find the value of the real number $a$.  
$(2)$ When $a=2$, find the maximum value of $f(x)$ on the interval $[0,\pi]$.  
$(3)$ If $f(x)\geqslant 0$ for all $x\in [0,\pi]$, find the range of values for the real number $a$.

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