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

(Ⅰ) If $x=0$ is an extremum point of $F(x)$, find the value of $a$.
(Ⅱ) When $a=1$, let $P(x_1, f(x_1))$ and $Q(x_2, g(x_2))$ ($x_1 > 0, x_2 > 0$), and suppose line segment $PQ$ is parallel to the x-axis, find the minimum distance between points $P$ and $Q$.
(Ⅲ) If when $x \geq 0$, the graph of the function $y=F(x)$ is always above the graph of $y=F(-x)$, find the range of values for the real number $a$.

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