
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 $a \in \mathbb{R}$ and the function $f(x) = e^x - ax$ (where $e = 2.71828\ldots$ is the base of the natural logarithm),  
(I) If the function $f(x)$ is decreasing in the interval $(-e, -1)$, find the range of values for $a$;  
(II) If the function $F(x) = f(x) - \left(e^x - 2ax + 2\ln x + a\right)$ has no zeros in the interval $\left(0, \frac{1}{2}\right)$, find the maximum value of $a$.

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