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

(I) When $a= \frac{1}{3}$, if there exist $x_{1}$, $x_{2}\in[0,+\infty)$ such that $f(x_{1})=g(x_{2})$, find the minimum value of $x_{2}-x_{1}$; if not, please explain why;

(II) If for $x\in[0,+\infty)$, $F(x)\geqslant F(-x)$ always holds, find the range of $a$.

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