
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 $f(x)=\frac{4x}{3x^{2}+3}(x\in(0,2)),g(x)=\frac{1}{2}x^{2}-\ln x-a$.
(1) Find the range of $f(x)$;
(2) If there exists $x\in[1,2]$ such that $g(x)=0$, find the range of $a$;
(3) For any $x_{1}\in(0,2)$, there always exists $x_{2}\in[1,2]$ such that $f(x_{1})=g(x_{2})$, find the range of $a$.

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