
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 function $f(x)=\lg (-x^{2}+2x+3)$ have an increasing interval $A$, and let the value range of $g(x)=\frac{1}{x^{2}-2x+2}$ be $B$, and let $C=\{ x| a-1\leqslant x\leqslant 2-a \}$.

$(1)$ Find $A\bigcap B$; 

$(2)$ If $A\bigcup C=A$, find the range of the real number $a$.

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