
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 vectors $$\overrightarrow {m} = (e^x, \ln x + k)$$ and $$\overrightarrow {n} = (1, f(x))$$, where $$\overrightarrow {m} \parallel \overrightarrow {n}$$ ($k$ is a constant, $e$ is the base of the natural logarithm), the curve $y=f(x)$ at the point $(1, f(x))$ has a tangent line perpendicular to the y-axis, and $F(x) = xe^xf'(x)$.  
(I) Find the value of $k$ and the monotonic intervals of $F(x)$;  
(II) Given the function $g(x) = \ln x - ax$ ($a > 0$), 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 values for the real number $a$.

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