
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 $f(x)=\frac{4x}{3x^2+3}$ $(x\in (0,2))$, $g(x)=\frac{1}{2}x^2-\ln x-a$.

$(1)$ If there exists $x\in [1,2]$ such that $g(x)=0$, find the range of values for $a$;

$(2)$ For every $x_1\in (0,2)$, there always exists $x_2\in [1,2]$ such that $f(x_1)=g(x_2)$, find the range of values for $a$.

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