
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 $f(x)=ax^{2}-2ax+b$ $(a > 0)$ has a maximum value of $3$ and a minimum value of $-1$ in the interval $[0,3]$.

$(1)$ Find the values of the real numbers $a$ and $b$;

$(2)$ Let $g(x)= \frac{f(x)}{x}$, if the inequality $g(3^{x})-k\cdot3^{x}\geqslant 0$ holds for all $x\in[-1,0)$, find the range of the real number $k$.

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