
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 $f(x)=ax^{2}-2x+1(a>0)$ and $g(x)=x+\frac{1}{x}$.  
$(1)$ When $a=3$, find the minimum and maximum values of the function $f(x)$ on the interval $[0,1]$.  
$(2)$ If for all $x_{1}\in (2,3)$, there always exists $x_{2}\in [1,2]$ such that the inequality $f(x_{1}) < g(x_{2})$ holds, find the range of real numbers for $a$.

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