
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
Let the function $f(x) = \ln x - e^{1-x}$, and $g(x) = a(x^2 - 1) - \frac{1}{x}$.
1. Determine the number of zeros of the function $y = f(x)$, and explain the reason;
2. Let $h(x) = g(x) - f(x) + \frac{e^x - ex}{xe^x}$, discuss the monotonicity of $h(x)$;
3. If $f(x) < g(x)$ holds true for $(1, +\infty)$, find the range of the real number $a$.

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