
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 that $a \in R$, the function $f(x) = e^x - ax$ (where $e = 2.71828...$ is the base of the natural logarithm).

(I) If the function $f(x)$ is a decreasing function in the interval $(-e, -1)$, find the range of values for $a$.

(II) If the function $F(x) = f(x) - (e^x - 2ax + 2 \ln x + a)$ has no zero points in the interval $(0, \frac{1}{2})$, find the maximum value of $a$.

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