
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
Let the function $f(x)=e^x+\sin x$, $g(x)=ax$ ($a\in \mathbb{R}$), and $F(x)=f(x)-g(x)$,

$(1)$ When $a=1$, let $P(x_1,f(x_1))$, $Q(x_2,g(x_2))$ ($x_1, x_2 \geqslant 0$), and the line segment $PQ$ is parallel to the $x$-axis. Find the minimum distance between points $P$ and $Q$;

$(2)$ If for $x \geqslant 0$, the graph of the function $y=F(x)$ is always above the graph of the function $y=F(-x)$, find the range of the real number $a$.

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