
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 functions $f(x)=\ln{x}$ and $g(x)=x+\frac{a}{x}$ with $a\in\mathbb{R}$.
1. Let $F(x)=f(x)+g(x)-x$. If the minimum value of $F(x)$ on the interval $[1,e]$ is $\frac{3}{2}$, find the value of the real number $a$.
2. If for $x\ge1$, the inequality $f(x)\le g(x)$ always holds, determine the range of values for the real number $a$.

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