
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 $f(x)=b\ln x$, $g(x)=ax^{2}-x$ $(a\in\mathbb{R})$  
$(1)$ If the curve of $f(x)$ and $g(x)$ have the same tangent line at their common point $A(1,0)$, find the values of the real numbers $a$, $b$;  
$(2)$ If $a > 0$, $b=1$, and the curves of $f(x)$ and $g(x)$ always have a common tangent line, find the minimum value of the positive number $a$.

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