
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
Let the domain of the function $f(x)= \sqrt {-x^{2}+2x+3}$ be the set $A$, and the range of the function $g(x)=x^{2}-x+1$, $x\in\mathbb{R}$, be the set $B$.
$(1)$ Find $A\cap B$;
$(2)$ If for any $x\in(0,+\infty)$, the inequality $g(x)\geqslant kx$ always holds, find the range of the real number $k$.

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