
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 the functions $f(x) = \ln x$ and $g(x) = \frac{t}{x} - \ln x$.  
(1) If $g(x) \le f(x)$ holds for all $x$, find the range of values for $t$.  
(2) Let $F(x) = f(x) - \frac{1}{e^{x}} + \frac{2}{e x}$. Determine whether the function $F(x)$ has any zeros, and if so, find the number of zeros; if not, provide a reason.

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