
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 $f(x)=x+\frac{a^2}{x}$ and $g(x)=x+\ln x$, where $a > 0$,

$(1)$ If $x=1$ is an extremum point of the function $h(x)=f(x)+g(x)$, find the value of the real number $a$;

$(2)$ If for any $x_1, x_2 \in [1,e]$ (where $e$ is the base of the natural logarithm), the inequality $f(x_1) \geqslant g(x_2)$ holds, find the range of values for the real number $a$.

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