
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 $f(x)=ax+\ln x$ $(a\in \mathbb{R})$ and $g(x)=x^{2}-2x+2$.
$(1)$ When $a=-\frac{1}{2}$, find the maximum and minimum values of the function $y=f(x)$ on the interval $[1,e]$.
$(2)$ If for any $x_{1}\in [-1,2]$, there exists $x_{2}\in (0,+\infty)$ such that $g(x_{1}) < f(x_{2})$, find the range of values for $a$.

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