
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 $f(x)= \frac {\ln x+k}{e^{x}}$ has its tangent line at point $(1,f(1))$ perpendicular to the $y$-axis, and $F(x)=xe^{x}f′(x)$  
$(1)$ Find the value of $k$ and the monotonic intervals of $F(x)$;  
$(2)$ Given the function $g(x)=-x^{2}+2ax$ ($a$ is a positive real number), if for any $x_{2} \in [0,1]$, there always exists $x_{1} \in (0,+\infty)$, such that $g(x_{2}) < F(x_{1})$, find the range of the real number $a$.

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