
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 $f(x)=e^{x}+\sin x,g(x)=ax$, and $F(x)=f(x)-g(x)$.

(1) Is $x=0$ an extreme point of $F(x)$? And find $a$.

(2) If $F(x)\geqslant 1$ holds for all $x\in[0,+\infty)$, find the range of values for the real number $a$.

(3) When $a=\frac{1}{3}$, there exist $x_{1},x_{2}\in [0,+\infty)$ such that $f(x_{1})-g(x_{2})$ holds. Find the minimum value of $(x_{2}-x_{1})$.

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