
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 function $f(x) = x - a\ln{x}$, $g(x) = -\frac{1+a}{x}$, where $a \in \mathbb{R}$.
1. Let the function $h(x) = f(x) - g(x)$. When $a > 0$, find the intervals of monotonicity for the function $h(x)$.
2. If there exists a point $x_0$ in $[1, e]$ ($e = 2.718\ldots$) such that $f(x_0) < g(x_0)$, find the range of values for $a$.

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