
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 $f(x)=ax+\frac{b}{x}(4a>b>0)$ passing through the point $(2,\frac{5}{2})$, and having a minimum value of $\sqrt{6}$ on $(0,+\infty)$.  
$(1)$ Find $a$ and $b$;  
$(2)$ For $x>0$, find the minimum distance from a point on the graph of $f(x)$ to $Q(2,4)$.

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